Question

Determine the equation of the parabola that opens up, has focus (−2,7), and a focal diameter of 24.

Answer 1

Answer:

$\sf{(x + 2)}^{2} = 24(y -1)$

Step-by-step explanation:

Given:

focus(h,k)= (−2,7)

Focal Diameter (Latus rectum) = 24

To find:

Equation of parabola= ?

Solution:

Focal Diameter also known as letus rectum is the segment which passes through Focus and intersects the Axis of parabola having end point at the wall of of parabola.

Focal Diameter = 24

$4p = 24 \\ p = \cancel\frac{ 24}{ 4} = 6 \\ \fbox{p = a = 6}$

Now to find the equation of parabola we need the vertex(h,v).

Vertex is a point on parabola which lies in same axis of symmetry at equidistant from the directrix and Focus and the distance is one fourth of letus rectum.

Now,

$\sf \: (h,v) = (h,k-p) \\ \sf \: (h,v)= (-2,7-6) \\\sf \: \fbox{ (h,v) = (-2,1)}$

Equation of parabola

$\sf{(x-h)}^{2} = 4a(y - v)$

Substituting the cordinate of vertex in above equation,

$\sf{(x - (-2)}^{2} = 4 \times ( 6)(y-1 ) \\ \sf{(x + 2)}^{2} = 24(y -1)$

This is the final answer.

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