Question

What is the equation of the parabola with a focus at (10,19) and a directrix of y= 22?

Answer 1

Equation of parabola is `x^2-20x+6y-23=0`

Explanation:

Here the directrix is a horizontal line `y=22`.

Since this line is perpendicular to the axis of symmetry, this is a regular parabola, where the x part is squared.

Now the distance of a point on parabola from focus at `(10,19)` is always equal to its between the vertex and the directrix should always be equal. Let this point be `(x,y)`.

Its distance from focus is `sqrt((x-10)^2+(y-19)^2)` and from directrix will be `|y-22|`

Hence, `(x-10)^2+(y-19)^2=(y-22)^2`

or `x^2-20x+100+y^2-38y+361=y^2-44y+484`

or `x^2-20x+6y+461-484=0`

or `x^2-20x+6y-23=0`