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The distance between the foci of a hyperbola is 2c. The distance formula to derive the equation equal to:
2a
2b
2c
The distance formula can be used to find the distance between:
The center and a vertex of a hyperbola
the center and a focus of a hyperbola
Two points on the minor axis of a hyperbola
If the distance between the foci of a hyperbola is 10 and a point on the hyperbola is (5,3), the equation for the distance using the distance formula is:
(x−5)2+(y−3)2=10\left(x-5\right)^2+\left(y-3\right)^2=10(x−5)2+(y−3)2=10
(x−5)2−(y−3)2=10\left(x-5\right)^2-\left(y-3\right)^2=10(x−5)2−(y−3)2=10
(x+5)2+(y−3)2=10\left(x+5\right)^2+\left(y-3\right)^2=10(x+5)2+(y−3)2=10
Given a hyperbola with foci at (0,2) and (0,−2) and a distance between the foci of 4, the equation of the hyperbola is:
x2−16y2=1x^2-16y^2=1x2−16y2=1
x2+16y2=1x^2+16y^2=1x2+16y2=1
For a hyperbola with foci at (0,3) and (0,−3), and a distance between the foci of 6, the equation of the hyperbola is:
9x2−16y2=139x^2-16y^2=139x2−16y2=13
6x2−25y2=16x^2-25y^2=16x2−25y2=1
9x2−16y2=09x^2-16y^2=09x2−16y2=0
When using the distance formula to find the distance between a point on a hyperbola and one of its foci, the result should be equal to:
The length of the major axis
the length of the minor axis
The distance between the foci
When deriving the equation of a hyperbola using foci and the distance formula, the term involving in the equation is typically squared because:
It makes the equation symmetric
It simplifies the calculations
The hyperbola can be oriented either horizontally or vertically
In the distance formula, the variables x1 and y1 represent:
The coordinates of the center of the hyperbola
the coordinates of one focus of the hyperbola
The coordinates of the other focus of the hyperbola
To derive the equation of a hyperbola using foci and the distance formula, the standard form should be of the type:
a2x2−b2y2=1a^2x^2-b^2y^2=1a2x2−b2y2=1
a2x2+b2y2=1a^2x^2+b^2y^2=1a2x2+b2y2=1
a2x2+b2y2=10a^2x^2+b^2y^2=10a2x2+b2y2=10
The distance formula is used to calculate the distance between:
The two foci of a hyperbola
the center and a vertex of a hyperbola
A point on a hyperbola and one of its foci
It is done.