What is the equation of the ellipse given in the example?
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4x² + 9y² = 36.
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What is the equation of the ellipse given in the example?
4x² + 9y² = 36.
What are the coordinates of one vertex of the ellipse?
(1, -2)
How can the position of a point be described in polar coordinates?
By considering the distance from the origin (r) and the angle (θ) with the positive x-axis.
What is the center of the circle?
C = (-4, 5).
What is the general equation of a circle?
𝑥 − ℎ² + 𝑦 − ℎ² = 𝑟².
What is the center of the ellipse?
The midpoint between the two foci F1 and F2.
What is the compound angle formula for cosine?
cos(α + θ) = cos(α)cos(θ) - sin(α)sin(θ)
What is the axis of symmetry in a parabola?
A line that divides the parabola into two mirror-image halves.
What is the vertex of a parabola?
The point where the parabola changes direction.
What are conic sections?
Curves that result from intersecting a right circular cone with a plane.
What is the general equation of an ellipse centered at C = (h, k)?
The equation is (x - h)²/a² + (y - k)²/b² = 1.
In the general equation of an ellipse, what do 'a' and 'b' represent?
'a' represents the semi-major axis and 'b' represents the semi-minor axis.
Why is the constant in the ellipse equation chosen as 2a?
To make the equation of the ellipse look 'nicer'.
What is the equation of the directrix for the parabola?
y = -a.
What are the four types of conic sections?
Circle, Ellipse, Hyperbola, Parabola.
What is the fixed point in a parabola called?
Focus.
How is the equation of the hyperbola derived from the original equation?
By completing the square.
What is the general vertex of a parabola?
The vertex can be at (h, k) instead of the origin (0, 0).
What is the standard equation of an ellipse?
x²/a² + y²/b² = 1.
What are the two representations of coordinate systems?
Cartesian coordinates (x, y) and Polar coordinates (r, θ).
What are the Cartesian coordinates of point B?
(1, -1)
What is the center of the hyperbola?
(2, 3).
What points does the ellipse pass through?
The points (a, 0), (-a, 0), (0, b), (0, -b) which are called vertices.
What are the coordinates of the foci of the ellipse?
F1 = -c, 0 and F2 = c, 0.
What does the equation y - ? = ±√(A/C)(x - ?) represent?
A pair of (unparallel) straight lines.
What are the fixed points in an ellipse called?
Foci (denoted as F1 and F2).
What is the defining property of points P in an ellipse?
P F1 + P F2 = constant = 2a.
What are the coordinates of point D in Cartesian form?
D ≈ (0.35, 1.97)
What happens to the equation when F' = 0?
It becomes A(x - ?)^2 + C(y - ?)^2 = 0, representing a pair of (unparallel) straight lines.
What happens when the x-axis and y-axis are rotated through a positive angle?
A new x' and y' axis is formed, creating a new coordinate system.
How is the coordinate P represented in the x, y coordinate system?
x = r cos(θ) + α, y = r sin(θ) + α.
What is the fixed distance from the center of a circle to any point on the circle called?
The fixed distance is called the radius and is denoted by r.
What is the focus of the parabola in the standard case?
F = (0, a) where a > 0.
What type of conic section is represented by the equation given?
Hyperbola.
What are the coordinates of the vertices of the hyperbola?
(2, 4) and (2, 2).
What is the value of 'a' for the hyperbola?
What are the coordinates of point P in the x', y' coordinate system?
x' = r cos(α), y' = r sin(α).
In which direction is the angle considered positive during the rotation of axes?
Anti-clockwise direction.
What is the relationship between tan(2θ) and the coefficients A, B, and C?
tan(2θ) = (B / (A - C)) if A ≠ C.
How can you determine if a conic section is a hyperbola from its equation?
By rearranging the equation into the standard form of a hyperbola.
What is the equation of the ellipse given in the example?
4x² + 9y² - 8x + 36y - 9 = 0.
What is the general equation of a conic section?
𝐴𝑥² + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
What are the equations used to describe the motion of a moving particle in physics?
The pair of equations: x = f(t) and y = g(t), where t is a parameter.
What is the transformation formula for y in the context of the rotation?
y = x' sin 30° + y' cos 30° = (1/2)x' + (√3/2)y'.
What is the general equation of a conic section?
A x² + Bxy + C y² + Dx + Ey + F = 0
What is a parabola?
The set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus).
What is the center of a circle called?
The center is called point C.
What are the coordinates of point C in Cartesian form?
C = (3/2, -3√3/2)
What do the variables h and k represent in the hyperbola equation?
They represent the coordinates of the center C = (h, k).
What transformation is applied to the x-y axes in the given text?
The axes are rotated by 30° in an anti-clockwise direction.
What is the equation of a parabola with a vertical axis of symmetry?
If the axis of symmetry is x = h, the equation is (x - h)² = 4a(y - k).
What is the initial equation after substituting the formulae?
5𝑥′² + 𝑦′² + 6𝑥′ + 𝑦′ − 𝑥′ + 𝑦′ + 5² − 𝑥′ + 𝑦′² − 18² + 2𝑥′ + 𝑦′ − 14 − 𝑥′ + 𝑦′ + 26 = 0.
What is the axis of symmetry for the parabola in Example 9?
The x-axis.
What is the equation of the given conic section?
xy = 1.
What is the general form of the equation for the parabola described?
y² = 4ax.
What is the final form of the equation derived from the simplification?
𝑥′² + 4𝑦′² − 2𝑥′ − 16𝑦′ + 13 = 0.
What type of conic section is represented when 𝐴𝐶 = 0?
Parabola.
What is the relationship between a, b, and c in an ellipse?
a > b > 0 and a > c.
What is the first step in converting Cartesian coordinates to polar coordinates?
Locate the points on the x-y plane.
What is the slope of the line segment PQ?
5/2
What are the vertices of the hyperbola?
A1 = (-a, 0) and A2 = (a, 0).
What coordinate system will be adopted in the remaining section of the lecture?
The Cartesian coordinates system.
What defines a circle in coordinate geometry?
A circle is a set of all points such that the distance between the point and a fixed point (the center) is always fixed.
Why is it important to rewrite the conic section equation into the form 𝑥−ℎ²/𝑝−𝑦−𝑘²/𝑞=1?
To address the existence of the term 𝐵𝑥𝑦.
What is the given equation of the conic section in the example?
13x² + 10xy + 13y² = 72.
What technique can be used to classify conic sections from their equations?
Completing the square technique.
What point does the circle pass through?
P = (-2, 1).
How can the equation of the ellipse be rewritten in standard form?
x²/9 + y²/4 = 1.
What is the coordinate of another vertex of the ellipse?
(3, -5/2)
What type of conic section is suggested by the graph?
An ellipse.
What is the significance of the coordinates (1, 2) in the context of the ellipse?
They represent the center of the ellipse.
What are the coordinates of the vertex (3, -5/2) related to?
It is one of the vertexes of the ellipse.
What is the standard form of the ellipse after rewriting?
(x - 1/2)²/7² + (y + 2)²/3² = 1.
How is the y-coordinate related to the x' and y' coordinates?
y = y'cos(θ) + x'sin(θ)
What does θ represent in polar coordinates?
The directed angle between OP and the positive x-axis.
What transformation is applied to the coordinate system?
The x-axis and y-axis are rotated by 45° in a clockwise direction.
What is the significance of rewriting conic section equations into standard form?
It allows for easy classification into circle, ellipse, parabola, or hyperbola.
Where are the foci located if the ellipse is oriented vertically?
At F1 = (0, c) and F2 = (0, -c) on the y-axis.
What technique is used to find the center and radius of the circle?
Completing the square.
What are the Cartesian coordinates of point A?
(3, 3)
What does the center of the ellipse (h, k) indicate?
It indicates the coordinates of the center of the ellipse.
How can one identify the conic section from the equation?
By sketching the graph of the conic section.
What coordinates are highlighted in red in the graph?
Coordinates in the x'y' plane.
How do you express the equation of line L1 in slope-intercept form?
From 3x - 2y + 5 = 0, we get y = (3/2)x + 5/2.
What is the derived equation of the ellipse after simplification?
a(x - c)^2 + y^2 = a^2 - xc.
What are the coordinates of point P through which line L passes?
(3, 4)
What is the equation of the parabola given in the example?
y² + ax - 4y + 3 = 0.
What equation is derived after substituting the transformation into the original equation?
4x'² + 2y' + 4 = 0.
What equation is represented by the coordinates given in the lecture note?
The equation of the ellipse.
What is the general equation of a conic section used for?
To classify and identify the conic section in the 2-D plane.
What condition must be met for A and C in the equation of conic sections?
If A > 0, then A and C must have the same sign (both positive or both negative).
What is the equation of the hyperbola when the foci are at F1 = (0, c) and F2 = (0, -c)?
The equation is given by y²/a² - x²/b² = 1, where b² = c² - a².
What conditions characterize a circle or ellipse?
A = C and B = 0
What is the 'x-coordinate' of a point P in Cartesian coordinates?
The directed distance from point P to the y-axis.
What is the result of B² - 4AC for the given conic section?
What cases are not considered in the classification of conic sections?
Degenerate cases.
What is the standard equation of a parabola that opens to the left?
y² = -4ax.
Where does line L cut the x-axis?
(5, 0)
What condition must be met for either A > 0 and C < 0 or A < 0 and C > 0?
If A*C < 0.
What is the fixed line in a parabola called?
Directrix.
What is the general form of the equation of a hyperbola?
The general form is (x - h)²/a² - (y - k)²/b² = 1 or (y - k)²/a² - (x - h)²/b² = 1.
What does the term 𝐵𝑥𝑦 indicate about a conic section?
It indicates that the conic section is being rotated from its 'standard position'.
What is the standard equation of a hyperbola with foci at (c, 0) and (-c, 0)?
The equation is x²/a² - y²/b² = 1.
What is the vertex of the parabola in Example 9?
The origin.
What is the first step to find the center and vertices of the hyperbola given the equation -x² + 4y² + 4x - 24y + 28 = 0?
Rearranging the equation into standard form.
What condition must be satisfied for b² in the ellipse equation?
b² = a² - c² > 0.
What are the Cartesian coordinates of point C?
(-2, -2)
What condition must be met for classifying conic sections?
At least one of 𝐴 or 𝐶 must be non-zero.
What is the slope of line L1?
The slope m1 of L1 is 3/2.
What does the variable 'a' represent in the context of the parabola?
The distance from the focus to the directrix.
What is the general form of the equation for a parabola given in the notes?
𝑥 ′ 2 = −1/4 2𝑦 ′ + 4.
What are the coordinates of the center of the ellipse?
(1/2, -2).
What do the variables r, α, and θ represent in the context of the formulas?
r is the radius, α is the angle, and θ is the rotation angle.
What does 2a represent in the context of an ellipse?
The sum of distances from any point on the ellipse to the two foci.
What is the general form of the equation after completing the square for conic sections?
A(x - ?)² + C(y - ?)² = F' where (F' ≠ 0).
In which direction is θ measured if it is less than 0?
Clockwise direction.
What type of conic section is represented by the equation (1/2)x'² - y'² = 1?
Hyperbola.
What condition indicates an ellipse?
B² - 4AC < 0
In the notation P = (a, b), what do 'a' and 'b' represent?
'a' is the x-coordinate and 'b' is the y-coordinate of point P.
What is the standard form of the equation for an ellipse?
x - h²/a² + y - k²/b² = 1.
What is a key technique for interchanging between Cartesian and Polar coordinates?
Using the relationships between the two systems.
What are the coordinates of point A in Cartesian form?
A = (1/√2, 1/√2)
What is the equation of the circle given in the example?
x² + y² + 8x - 10y - 8 = 0.
What is the equation of an ellipse when the foci lie on the y-axis?
x²/b² + y²/a² = 1, where b² = a² - c².
What is the equation of a circle with its center at the origin?
x² + y² = r².
What happens to the equation of the ellipse when a = b?
It becomes x² + y² = a², which is the equation of a circle.
How do you convert polar coordinates to Cartesian coordinates?
Use the formulas x = r cos(θ) and y = r sin(θ).
What is the first step in classifying a conic section given its equation?
Rewrite the equation into the standard form.
What is the relationship between a, b, and c in the context of a hyperbola?
b² = c² - a², where a, b > 0.
What technique is used to rewrite the equation of the ellipse?
Completing the square.
What is the general form of the conic section equation after rotation?
A'x'^2 + C'y'^2 + D'x' + E'y' + F' = 0.
How is the x-coordinate related to the x' and y' coordinates?
x = x'cos(θ) - y'sin(θ)
What is the relationship defined for a hyperbola involving its foci?
PF1 - PF2 = 2a.
What transformation is applied to the coordinate system?
The x-axis and y-axis are rotated by 45° in an anti-clockwise direction.
What is the first step in the classification procedure of conic sections?
Transform the equation using the transformation formulas.
What happens to the parabola when a = 1?
The equation becomes y² - 4y + x + 3 = 0.
What is the significance of the coordinates (9/2, -2) in the context of the ellipse?
It is another vertex of the ellipse.
What are the semi-major and semi-minor axes of the ellipse?
Semi-major axis = 7, Semi-minor axis = 3.
What type of conic section is represented when 𝐴𝐶 > 0?
Circle or Ellipse.
What is the transformation formula for y in the new coordinates?
y = x' sin(-45°) + y' cos(-45°) = (1/2) - x' + y'.
What do the values f(b) and g(b) represent in parametric equations?
They represent the coordinates (x, y) at the endpoint of the motion.
What is the equation for 𝐶 ′ in terms of 𝐴, 𝐵, 𝐶, and 𝜃?
𝐶 ′ = 𝐴 sin²(𝜃) - 𝐵 sin(𝜃) cos(𝜃) + 𝐶 cos²(𝜃).
What transformation is needed to rewrite the conic section equation?
Transform into the form A'x'² + C'y'² + D'x' + E'y' + F' = 0 using rotation of axes.
What does the equation become after choosing the appropriate 𝜃?
𝐴'𝑥'² + 𝐶'𝑦'² + 𝐷'𝑥' + 𝐸'𝑦' + 𝐹' = 0.
What is the general form of the equation after completing the square for a hyperbola?
A(x - ?)^2 - C(y - ?)^2 = F' where (F' ≠ 0).
What is the standard equation of an ellipse centered at the origin?
x²/a² + y²/b² = 1.
What are the coordinates of point B in Cartesian form?
B = (-√3, 1)
What is the general form of a conic section in 2-D?
𝐴𝑥² + 𝐵𝑥𝑦 + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0, where 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹 are constants.
What type of conic section is described in the lecture note?
A standard ellipse.
What is the center of the circle in the example?
C = (2, 3).
What is the center of the ellipse in the x'y' plane?
(1, 2).
What is the equation of the circle with center (1, -1) and radius 2?
The equation is (x - 1)² + (y + 1)² = 2².
What does cos(2θ) equal if A = C?
cos(2θ) = 0.
What are the four types of conic sections?
Circle, Ellipse, Parabola, and Hyperbola.
What is the standard form equation of a hyperbola?
x²/a² - y²/b² = 1.
What type of conic section is represented by the equation 4x² - 16x + 25y² - 84 = 0?
This equation can be classified after completing the square.
What is a useful technique for identifying conic sections in the 2-D plane?
Rotation of Axes.
What happens when you take the square of both sides in the ellipse equation derivation?
You expand the terms to derive the equation of the ellipse.
What are the two types of coordinate systems used in 2-D?
Cartesian coordinates and another unspecified type.
What transformation is applied to the equation to express it in a different form?
Completing the square.
What is the value of c for the foci of the ellipse?
c = 5.
In which direction is θ measured if it is greater than 0?
Anti-clockwise direction.
What is the final equation after completing the square?
𝑥′ − 1/2)²/2² − (𝑦′ − 2)²/1² = 1.
What is the standard equation of a parabola that opens downwards?
x² = -4ay.
What is the general form of the equation for a conic section?
𝐴𝑥² + 𝐵𝑥𝑦 + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
What is the significance of the coefficients in the parabola equations?
They determine the orientation and position of the parabola.
What is the equation of line L1?
3x - 2y + 5 = 0
What happens to the equations of parabolas when x and y are interchanged?
The parabola's orientation changes.
What occurs in the degenerate case when F' = 0?
The conic section becomes a single point.
What is the general form of a conic section equation?
Ax² + Cy² + Dx + Ey + F = 0.
What does the variable 'r' represent in the equation of a circle?
The radius of the circle.
What is the equation of a parabola with a horizontal axis of symmetry?
If the axis of symmetry is y = k, the equation is (y - k)² = 4a(x - h).
How is the radius of the circle calculated?
r = CP = √((2 - (-2))^2 + (3 - 1)^2) = √20.
What condition must be satisfied for a point P(x, y) to lie on the parabola?
PQ = PF, where Q is the foot of the perpendicular from P to the directrix.
What does the parameter 't' typically represent in parametric equations?
Time.
What is the equation of the circle derived in the example?
x^2 + y^2 - 4x - 6y - 7 = 0.
What is the general equation of a hyperbola?
𝑥 − ℎ²/a² − 𝑦 − 𝑘²/b² = 1 or 𝑦 − 𝑘²/a² − 𝑥 − ℎ²/b² = 1.
Through which point does the parabola pass?
(6, 3).
If lines L and L1 are perpendicular, what is the relationship between their slopes?
m × m1 = -1.
What condition indicates that the conic section is an ellipse or circle?
A'C' > 0.
What value of 'a' is calculated for the parabola?
3/8.
What does b² represent in the context of hyperbolas?
b² = c² - a², where c > a.
What is the significance of the values f(a) and g(a) in parametric equations?
They represent the coordinates (x, y) at the starting point of the motion.
What is the equation of line L derived from its slope?
The equation is 2x + 3y - 10 = 0.
What are the asymptotes of the hyperbola?
The lines y = (b/a)x and y = -(b/a)x.
What is the significance of the values a and b in the hyperbola equation?
a represents the distance from the center to the vertices, and b represents the distance related to the asymptotes.
What are the coordinates of the foci of the hyperbola?
The foci are F1 = (0, c) and F2 = (0, -c).
What trigonometric identity is used in the equation for 𝐵 ′?
2 sin(𝜃) cos(𝜃) = sin(2𝜃) and cos²(𝜃) − sin²(𝜃) = cos(2𝜃).
What discriminant is used for classifying conic sections?
B² - 4AC.
What is the relationship that defines an ellipse in terms of its foci?
PF1 + PF2 = 2a.
What is the radius of the circle?
What is the compound angle formula for sine?
sin(α + θ) = sin(α)cos(θ) + cos(α)sin(θ)
What is the final form of the circle's equation after completing the square?
(x + 4)² + (y - 5)² = 49.
What are the coordinates of the foci of the ellipse?
(c, 0) and (-c, 0).
What is the range for the angle θ in polar coordinates?
−180° < θ ≤ 180°
What is the standard form of the equation after completing the square for 4x² - 16x + 25y² - 84 = 0?
4(x - 2)² + 25y² = 100.
What is the center of the hyperbola?
The origin (mid-point of the foci F1 and F2).
What is the significance of '2a' in the ellipse equation?
It represents the constant sum of distances from any point on the ellipse to the foci.
What is the axis of symmetry for the standard parabola?
The y-axis.
What is the expanded form of the equation of the circle with center (1, -1) and radius 2?
The expanded form is x² + y² - 2x + 2y = 0.
What is the equation of line PQ?
5x - 2y - 7 = 0
What is the effect of setting a = 0 in the parabola's equation?
The equation changes, affecting the shape and position of the parabola.
What is the center of the hyperbola described?
The center is at the origin (0, 0).
What is the condition for an ellipse in terms of 𝐴 and 𝐶?
𝐴 must not equal 𝐶.
What happens when A ≠ C in the conic section equation?
The equation represents an ellipse.
What is the equation used to solve for 𝐵 ′?
𝐵 ′ = −2𝐴 cos(𝜃) sin(𝜃) + 𝐵 cos²(𝜃) − sin²(𝜃) + 2𝐶 sin(𝜃) cos(𝜃) = 0.
What is the classification technique for conic sections using rotation of axes?
Using the equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.
What is the derived equation of the parabola with focus at (0, a) and directrix y = -a?
x² = 4ay.
What is the significance of the equation (x - h)²/a² + (y - k)²/b² = 1?
It describes the shape and position of an ellipse in a coordinate system.
How can the equation of a circle be expressed to find its center and radius?
In the form (x - h)² + (y - k)² = r².
What are the coordinates of the vertices of the ellipse?
(3, 0), (-3, 0), (0, 2), (0, -2).
What does 2a represent in the context of a hyperbola?
It is the difference between the distances from any point on the hyperbola to the two foci.
What is the given equation of the conic section?
5x² + 6xy + 5y² - 18x - 14y + 26 = 0.
What are the coordinates of point Q through which line L passes?
(1, -1)
What is the value of the radius squared (r^2) in the equation?
r^2 = 20.
How can the equations of conic sections be expressed in a general form?
𝐴𝑥² + 𝐶𝑦² + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0.
How is the position of a point described in Cartesian coordinates?
By the directed distances from the point to the x-axis and y-axis.
What is the vertex of the parabola when a = 1?
The vertex is at (h, k) = (1, 2).
What is the final equation of the parabola?
y² = (3/2)x.
What condition must be satisfied for b² in hyperbolas?
b² must be greater than 0 (b² > 0).
What is the first step to find the equation of the conic section in x'y' coordinates?
Apply the transformation formulas for x and y to the original conic equation.
What is the relationship between a, b, and c in the hyperbola equation?
b² = c² - a².
What condition should be chosen for 𝜃 to simplify the conic section equation?
Choose 𝜃 such that 𝐵'𝑥'𝑦' = 0 or 𝐵' = 0.
What is the final simplified equation derived from the conditions?
𝐴 − 𝐶 sin²(𝜃) = 𝐵 cos²(𝜃).
What is the general equation of an ellipse?
𝑥 − ℎ²/a² + 𝑦 − 𝑘²/b² = 1.
What are the relations between polar coordinates (r, θ) and Cartesian coordinates (x, y)?
x = r cos(θ), y = r sin(θ), r = x² + y², tan(θ) = y/x.
What does the equation simplify to after combining like terms?
2𝑥′² + 8𝑦′² − 4𝑥′ − 32𝑦′ + 26 = 0.
How can the general equation of a parabola be derived from the standard form?
By replacing x with (x - h) and y with (y - k) in the standard form.
What does r represent in polar coordinates?
The distance between the point P and the origin.
What visual technique is used to identify the conic section?
Rotating the viewpoint by a certain degree.
What condition characterizes a parabola in conic sections?
B² - 4AC = 0
What are the coordinates of the foci for the hyperbola in standard form?
F1 = (-c, 0) and F2 = (c, 0).
What type of conic section is represented when 𝐴𝐶 < 0?
Hyperbola.
What does the vertex of the parabola represent in the given equations?
The vertex is at the point (ℎ, 𝑘) = (0, -2).
How do you identify the type of conic section?
By calculating B² - 4AC.
What are the coordinates of the vertices of the hyperbola?
The vertices are A1 = (0, -a) and A2 = (0, a).
What is the condition for a circle in terms of 𝐴 and 𝐶?
𝐴 must equal 𝐶.
How is A' calculated in the transformation process?
A' = A cos² θ + B cos θ sin θ + C sin² θ.
What is the vertex form of a parabola derived from the equations?
The vertex form is represented as 𝑦 = 𝑎(𝑥 - ℎ)² + 𝑘.
What is the formula to find the angle of rotation for the conic section?
tan(2θ) = B / (A - C).
What is the transformation formula for x in the context of the rotation?
x = x' cos 30° - y' sin 30° = (√3/2)x' - (1/2)y'.
What is the general equation of a circle with center at C(h, k)?
(x - h)² + (y - k)² = r².
What is the general equation of a parabola?
𝑥 − ℎ² = 4𝑎(𝑦 − 𝑘) or 𝑦 − 𝑘² = 4𝑎(𝑥 − ℎ).
What is the general form of the conic section equation discussed?
Ax² + Bxy + Cy² + Dx + Ey + F = 0 (for B ≠ 0).
In the equation (x - h)² + (y - k)² = r², what do 'h' and 'k' represent?
The x and y coordinates of the center of the circle.
What condition indicates that the conic section is a parabola?
A'C' = 0.
What are the functions involved in the parametric equations?
f(t) and g(t).
What is the equation of the given conic section?
3x² + 2√3xy + y² - x + 3y + 4 = 0.
What is the range of the parameter 't' in the context of parametric equations?
t ∈ [a, b].
What is the slope of line L if the slope of line L1 is 3/2?
The slope m of line L is -2/3.
What shape does the parabola take in the 𝑥 ′ 𝑦 ′ - plane?
It is an inverted U-shape curve.
What type of conic section is represented when A = C?
The equation represents a circle.
How is the vertex form of the parabola derived in the example?
By completing the square on the equation.
What is the formula for B' in the transformed equation?
B' = -2A cos θ sin θ + B cos² θ - sin² θ + 2C sin θ cos θ.
What does the equation become when 𝐵 ′ = 0?
𝐴 ′ x ′² + 𝐶 ′ y ′² + 𝐷 ′ x ′ + 𝐸 ′ y ′ + 𝐹 ′ = 0.
What are the two basic coordinate systems?
Cartesian coordinates (xy-coordinate) and Polar coordinates (rθ-coordinate).
What should you know about the standard equations of conic sections?
You should know the standard equations for Circle, Ellipse, Parabola, and Hyperbola.
What is the center of the circle given by the equation x + 3² + y - 1² = 7?
The center is C = (-3, 1).
Where do the foci of the ellipse lie?
On the x-axis.
What is the vertex of the standard parabola?
The origin O = (0, 0).
What is the radius of the circle given by the equation x + 3² + y - 1² = 7?
The radius is √7.
What are the transformation formulas for x and y in the new coordinate system?
x = (1/√2)x' - (1/√2)y', y = (1/√2)x' + (1/√2)y'.
What transformation formulas are used to transform the conic section equation?
x = x' cos θ - y' sin θ and y = x' sin θ + y' cos θ.
What is the resulting equation after substituting the transformation into the conic section equation?
(1/2)x'² - y'² = 1.
What condition indicates a hyperbola?
B² - 4AC > 0
What is the 'y-coordinate' of a point P in Cartesian coordinates?
The directed distance from point P to the x-axis.
What is the equation of a circle derived from the condition A = C?
x - ?² + y - ?² = r².
What condition must be met to simplify the equation to 𝐵 ′ = 0?
Choose 𝜃 such that 𝐵 ′ = 0.
What is the calculated angle of rotation (θ) for the given conic section?
30°.
How can you identify a conic section?
Using the completing square technique.
What is the transformation formula for x in the new coordinates?
x = x' cos(-45°) - y' sin(-45°) = (1/2)x' + y'.
What condition indicates that the conic section is a hyperbola?
A'C' < 0.
What is the alternative form of the conic section equation after transformation?
A'x'² + B'x'y' + C'y'² + D'x' + E'y' + F' = 0.
What is the standard equation of a parabola that opens to the right?
y² = 4ax.
What type of conic section is identified from the equation?
Parabola.
What transformation formulas are used to rewrite the conic section equation?
𝑥 = 𝑥' cos 𝜃 − 𝑦' sin 𝜃 and 𝑦 = 𝑥' sin 𝜃 + 𝑦' cos 𝜃.
What are the coordinates of points B1 and B2 in relation to the hyperbola?
B1 = (-b, 0) and B2 = (b, 0).
What are the four conic sections?
Circle, Ellipse, Parabola, Hyperbola.
What are the coordinates of the foci of the ellipse?
F₁ = (-5, 0) and F₂ = (5, 0).
What happens to the graph of the hyperbola as x and y get larger?
The graph approaches the asymptotes y = (b/a)x and y = -(b/a)x.
What is the condition for line L to be perpendicular to line L1?
The product of their slopes must equal -1.
What is the first step to identify a conic section from its equation?
Use the transformation formula to rewrite the equation.
What is the significance of the expression B² - 4AC?
It helps classify the type of conic section.
What does 'a' represent in the equations of parabolas?
The distance from the vertex to the focus or directrix.
What is the main focus of Lecture Note 1 in MA1200?
Coordinate Geometry and Conic Sections.
How can one identify the conic section after rewriting the equation?
By using the results outlined in the reference material (P.39).
