Question

Which is the equation of a parabola with a directrix at y = −3 and a focus at (5, 3)? y = one twelfth(x − 5^)2

Answer 1

The equation of a parabola with a directrix at y = -3 and a focus at (5 , 3) is y = one twelfth (x - 5)² ⇒ 1st answer

Step-by-step explanation:

The 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 (5 , 3)

- Compare it with the 2nd rule above

∴ h = 5

∴ k + p = 3 ⇒ (1)

∵ The directrix is at y = -3

- By using the 3rd rule above

∴ k - p = -3 ⇒ (2)

Solve the system of equations to find k and p

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

∴ 2k = 0

- Divide both sides by 2

∴ k = 0

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

∵ 0 + p = 3

∴ p = 3

Substitute the values of h , k , and p in the form of the equation above

∵ (x - 5)² = 4(3)(y - 0)

∴ (x - 5)² = 12 y

- Divide both sides by 12

∴ $\frac{1}{12}$ (x - 5)² = y

- Switch the two sides

∴ y =  $\frac{1}{12}$ (x - 5)²

The equation of a parabola with a directrix at y = -3 and a focus at (5 , 3) is y =  $\frac{1}{12}$ (x - 5)²

Learn more:

you can learn more about the quadratic equations in brainly.com/question/8054589

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