Find the equation of the parabola with focus | vedclass

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Question

(A) Let the vertex be $A(4, -1)$ and the focus be $S(4, -3)$.The axis of the parabola is the line passing through the focus and the vertex,which is $x

Vedclass · Accepted Answer

$x^{2} - 8x + 8y + 24 = 0$

Vedclass · Answer

(A) Let the vertex be $A(4, -1)$ and the focus be $S(4, -3)$.The axis of the parabola is the line passing through the focus and the vertex,which is $x = 4$.Since the vertex $A(4, -1)$ is at a distance $a = |-1 - (-3)| = 2$ from the focus $S(4, -3)$,and the parabola opens downwards (as the focus is below the vertex),the equation of the parabola is of the form $(x - h)^{2} = -4a(y - k)$,where $(h, k)$ is the vertex.Here,$(h, k) = (4, -1)$ and $a = 2$.Substituting these values,we get:$(x - 4)^{2} = -4(2)(y - (-1))$$(x - 4)^{2} = -8(y + 1)$$x^{2} - 8x + 16 = -8y - 8$$x^{2} - 8x + 8y + 24 = 0$.

Find the equation of the parabola with focus $(4, -3)$ and vertex $(4, -1)$.

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