What is the equation of a parabola with (4,6) as its focus and y = 2 as its directrix

The equation of the parabola is (x - 4)² = 8(y - 4)

Step-by-step explanation:

* Lets revise the equation of a parabola

- If the equation is in the form (x − h)² = 4p(y − k), then:  

• Use the given equation to identify h and k for the vertex, (h , k)

• Use the value of h to determine the axis of symmetry, x = h

• Use h , k and p to find the coordinates of the focus, (h , k + p)  

• Use k and p to find the equation of the directrix, y = k − p

* Now lets solve the problem

∵ The directrix is y = 2

∴ The equation is (x - h)² = 4p(y - k)

∴ The focus is (h , k + p)

∵ The focus is (4 , 6)

∴ h = 4

∵ k + p = 6 ⇒ (1)

∵ The directrix is y = k - p

∴ k - p = 2 ⇒ (2)

* Add (1) and(2) to find k

∴ 2k = 8 ⇒ ÷ 2 for both sides

∴ k = 4

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

∵ 4 + p = 6 ⇒ subtract 4 from both sides

∴ p = 2

* Now lets write the equation

∴ (x - 4)² = 4(2)(y - 4) ⇒ simplify

∴ (x - 4)² = 8(y - 4)

* The equation of the parabola is (x - 4)² = 8(y - 4)

step-by-step explanation:

∆abc has the points a(1, 7), b(-2, 2), and c(4, 2) as its vertices. formed with the point d(1, 2) as its third vertex, then ∆abd is triangle.