Write the standard form of the equation of the parabola with focus (0,-2) and its vertex at the origin.
A) $y ^ { 2 } = 8 y$
B) $x ^ { 2 } = 8 x$
C) $y ^ { 2 } = - 8 x$
D) $x ^ { 2 } = - 8 y$
E) $x ^ { 2 } = \frac { 1 } { 8 } y$

Question

Quizplus · Accepted Answer

The standard form of the equation of a parabola with vertex at the origin and focus at (0, p) or (p, 0) is given by:
$$4p(y^2-x^2)=x^2+y^2$$
In this case, the vertex is at the origin, so p = -2 (since the focus is at (0, -2)). Substituting p = -2, we get:
$$4(-2)(y^2-x^2)=x^2+y^2$$
Simplifying, we get:
$$-8y^2+8x^2=x^2+y^2$$
$$7x^2+9y^2=0$$
Dividing both sides by 9, we get:
$$\frac{x^2}{-\frac{9}{7}}+\frac{y^2}{\frac{1}{7}}=1$$
Therefore, the standard form of the equation of the parabola is:
$$x^2 = -8y$$ which is option D.

Write the Standard Form of the Equation of the Parabola y2=8yy ^ { 2 } = 8 yy2=8y

Write the Standard Form of the Equation of the Parabola $y ^ { 2 } = 8 y$