Conic Sections Flashcards
What is the standard form of the equation for a parabola that opens to the right?
y² = 4ax
Fill in the blank: In the equation y² = 4ax, the value of ‘a’ represents the __________ of the parabola.
distance from the vertex to the focus
What is the standard form of the equation for a parabola that opens to the left?
y² = -4ax
What is the standard form of the equation for a parabola that opens to downward?
x² = -4ay
What is the standard form of the equation for a parabola that opens to upward?
x² = 4ay
For a parabola distance of focus from vertex is equal to distance of vertex from ______
Directrix
Equation of ellipse
x²/a² + y²/b² = 1
OR
x = a cosθ
y = b sinθ
__________________________
(x-h)²/a² + (y-k)²/b² = 1
OR
x = h + a cosθ
y = k + b sinθ
Standard form of hyperbola
x²/a² - y²/b² = 1
Equation for eccentricity of hyperbola
b² = a² (e² - 1)
Equation of eccentricity for ellipse
b² = a² (1- e²)
Distance from the center to the focus of an ellipse
ae = √ (a² - b²)
Coordinates of focus for a hyperbola / ellipse (horizontal)
(±ae, 0)
What is ‘a’ in eqn of ellipse
Semi major axis
The semi-major axis is half of the longest length of the ellipse, going through its center and both foci
What is ‘b’ in eqn of ellipse
Semi minor axis
It’s the shortest radius of the ellipse. Perpendicular to the semi-major axis
Distance between directrix of ellipse
2a/e
a: major axis
e: eccentricity
Condition for a line y = mx + c to touch the parabola y² = 4ax is
c= a/m
c: constant in the eqn of line
a: semi major axis
m: slope of line
Equation for focus of ellipse
ae = √(a² -b²)
Equation for focus of hyperbola
ae = √(a² + b²)
Coordinates of focus for a hyperbola / ellipse (vertical)
(0,±ae)
What is ‘a’ in the eqn of hyperbola
Semi major axis
it is the distance from the center to either vertex of the hyperbola
Eqn of parabola with axis y= k, vertex (h,k)
(y-k)² = 4a(x - h)
Eqn of parabola with axis, x=h, vertex (h,k)
(x-h)² = 4a(y-k)
Length of latus rectum of ellipse
2b²/a
Length of major axis of ellipse
2a
Length of minor axis of ellipse
2b
Distance between focii of an ellipse
2ae
Conditions for a line y=mx+c to be a tangent to the ellipse x²/a² + y²/b² = 1
- c² =a²m² + b²
- a²x/x1 - b²y/y1 = a² - b²
Conditions for a line y=mx+c to be a tangent to the hyperbola x²/a² - y²/b² = 1
