Find the area of triangle formed by the lines joining the vertex of the parabola x²=12y to the end of it's latus ractum

Question

Solve rhe question

 

Answer 1

Consider this variation of the vertex form of a parabola:

 

(x - h)² = 4p*(y - k)

 

Where (h,k) are the coordinates of the vertex and p is the distance from the vertex to the focus of the parabola. Since the latus rectum of a parabola passes through the focus, p is also the distance from the vertex to the latus rectum. The length of the latus rectum is found by multiplying p by 4, or 4p.

 

So, the triangle formed by connecting the vertex to the ends of the latus rectum has a height, p and a base, 4p. The area of a triangle is A = (1/2)*base*height.

 

Comparing the vertex form with the equation in the problem gives you:

 

(h,k) = (0,0)

4p = 12

 

Then, substitute your values for the base and height in the area formula to get:

 

A = (1/2)*(12)*(3) = 18