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(x-4)^2 =-24(y+1)

A parabola can be drawn given a focus of (4,-7) and a directrix y =5 . Write the equation of the parabola in any form.

  • Parabola with focus(4,-7) and directrix y =5

 

Given that the focus is at (4,-7) and the directrix is at  y=5.

The detailed article, parabolic equations Parabola Equation.Types of parabolic equations. can be reviewed for concepts on solving this problem step by step.

Let us follow the steps in the article to understand easy.

Step1: Firstly, Identify the orientation of the focus point coordinate and the directrix line whether it is a  horizontal or a vertical line.

  • The directrix line is the Horizontal dotted line y=5.and the focus is located below the  directrix  line.

Step2: Find the vertex (h,k) by finding the mid-point of the focus point and the directrix line.

  • The “y” coordinate of the focus point (4,-7)  is -7.So the vertical distance between the focus and the directrix is  7+5 = 12. Therefore the midpoint will be at a distance of  6  units from the focus. Hence  (4  , -7+6) = (4 ,-1) is the  vertex.

Step2:Get the focal length “a” from the vertex calculated and the focal point.

  • The focal distance we just found as 6. This is the  half way distance  between the focus and directrix y=5

Step4: If the directrix of the form Y= Constant number,( A horizontal line) then the parabola equation would be

1)(X-h)^2 = 4a(Y-k) (If the  focus point is above the directrix  line)

2)(X-h)^2=-4a(Y-k) (If the  focus point is below the directrix  line )

  • Since the focus point is below the directrix line, the vertex (h,k) is (4,-1) and the focal length is a= 6. The equation of the  parabola would be (X-h)^2=-4ay = (x- 4)^2 =-4*6*(y- (-1))

Therefore, the  final solution would be

 (x- 4)^2 =-24(y+1)

(x-4)^2 =-24(y+1)