What is the equation of a parabola with a directrix of y=2 and a focus point of 0,-2

Any point, (x0,y0)(x0,y0) on the parabola satisfies the definition of parabola, so there are two distances to calculate:

To find the equation of the parabola, equate these two expressions and solve for y0y0 .

Find the equation of the parabola in the example above.

Distance between the point (x0,y0)(x0,y0) and (a,b)(a,b) :

(x0−a)2+(y0−b)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√(x0−a)2+(y0−b)2

Distance between point (x0,y0)(x0,y0) and the line y=cy=c :

∣∣y0−c∣∣| y0−c |

(Here, the distance between the point and horizontal line is difference of their yy -coordinates.)

Equate the two expressions.

(x0−a)2+(y0−b)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=∣∣y0−c∣∣(x0−a)2+(y0−b)2=| y0−c |

Square both sides.

(x0−a)2+(y0−b)2=(y0−c)2(x0−a)2+(y0−b)2=(y0−c)2

Expand the expression in y0y0 on both sides and simplify.

(x0−a)2+b2−c2=2(b−c)y0(x0−a)2+b2−c2=2(b−c)y0

This equation in (x0,y0)(x0,y0) is true for all other values on the parabola and hence we can rewrite with (x,y)(x,y) .

Therefore, the equation of the parabola with focus (a,b)(a,b) and directrix y=cy=c is

(x−a)2+b2−c2=2(b−c)y

step-by-step explanation:

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step-by-step explanation: