Question

Write the equation for a parabola with a focus at (6, 7) and a directrix at x = 1.

Answer 1

ANSWER

$\text{ The equation of the parabola is; \lparen y - 7\rparen}^2\text{ = 10\lparen x - 3.5\rparen}$

EXPLANATION

Given that;

The focus of the parabola is (6, 7)

The directrix is 1

Follow the steps below to find the equation of the parabola

Step 1; Write the general formula of the parabola equation

$\text{ \lparen y - k\rparen}^2\text{ = 4p\lparen x - h\rparen}^$

Recall, that the vertex of the parabola midway between the focus and directrix

Hence, h can be calculated below has

$\begin{gathered} \text{ h = }\frac{\text{ k}}{\text{ 2}} \\ \\ \text{ h = }\frac{7}{2} \\ \text{ h = 3.5} \end{gathered}$

Also, p is the distance of vertex to directrix

p = 6 - 3.5

p = 2.5

Step 2; Substitute the calculated data into the formula in step 1

$\begin{gathered} \text{ \lparen y - 7\rparen}^2\text{ = 4}\times2.5\text{ \lparen x - 3.5\rparen} \\ \text{ \lparen y - 7\rparen}^2\text{ = 10\lparen x - 3.5\rparen} \end{gathered}$