Question

Write the equation for the parabola with a focus at (6,-4) and a directix at y=-7.

Answer 1

The equation of the parabola is (x - 6)² = 6(y + 5.5)

Step-by-step explanation:

The standard form of the equation of the parabola is  (x - h)² = 4p(y - k), where

• The vertex of the parabola is (h , k)
• The focus is (h , k + p)
• The directrix is at y = k - p  

∵ The focus of the parabola is at (6 , -4)

- the coordinates of the focus are (h , k + p)

∴ h = 6

∴ k + p = -4 ⇒ (1)

∵ The directrix of the parabola is at y = -7

- The directrix is at y = k - p

∴ k - p = -7 ⇒ (2)

Solve the system of equations to find k and p

Add equations (1) and (2) to eliminate p

∴ 2k = -11

- divide both sides

∴ k = -5.5

- Substitute the value of k in equation (1) to find p

∵ -5.5 + p = -4

- Add 5.5. to bith sides

∴ p = 1.5

∵ The form of the equation is (x - h)² = 4p(y - k)

- Substitute the values of h, k, and p in the form

∴ (x - 6)² = 4(1.5)(y - -5.5)

∴ (x - 6)² = 6(y + 5.5)

The equation of the parabola is (x - 6)² = 6(y + 5.5)

Learn more:

You can learn more about the parabola in brainly.com/question/9390381

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