Question

Write an equation for the set of all points in the plane equidistant from (0, – 1) and y= – 5. Write your answer in vertex form. Simplify any fractions

Answer 1

Let a point P = $(x, y)$ .

Given,

A = (0, -1) and

Equation of a line is $\space y + 5 =0$

PA = $\sqrt {(x_2-x_1)^2 + (y_2 -y_1)^2} = \sqrt {(0-x)^2 + (-1-y)^2}$

PA = $\sqrt {x^2 + y^2 + 2y+ 1}$

Perpendicular Distance from the point $(x,y)$ to the Line L: $y +5 = 0$

PL = $|\frac{x(0) +y(1) +5}{\sqrt {0+1}} | = |{y+5}|$

Given, PA = PL

$$PA^2 = PL^2$$

$$x^2+y^2 +2y +1 = (y+5)^2$$

$$x^2 +y^2 + 2y +1 = y^2 +10y + 25$$

$x^2 = 8y+24$

$$x^2 = 8 (y +3)$$

The equation of set of points in vertex form is, $x^2 = 8(y+3)$