Question

Using a directrix of y = −3 and a focus of (2, 1), what quadratic function is created?

Answer 1

Equation of directrix is y = 3 and co-ordinate of focus is (2, 1)
Let (x , y) is the point on the parabola [ ∵ quadratic equation is known as parabolic equation]
We know, distance between point on the parabola and focus is equal to distance between point on the parabola and directrix.
∴ distance between (x , y) and (2,1) = distance between y = 3 and (x ,y)
$\sqrt{(x -2)^2+(y-1)^2}=|y-3|$
Squaring both sides,
(x -2)² + (y -1)² = (y - 3)²
⇒ x² - 4x + 4 = y² - 6y + 9 - y² -2y - 1
⇒ x² - 4x + 4 = -4y + 8
⇒ x² - 4x + 4y - 4 = 0
⇒y = -x²/4 + x + 1

Hence, quadratic equation is -x²/4 + x + 1