back to analytical geometry video lessons

Parabola Example 6

Question:

A parabola has the equation y2 = 4x. Points A and B lie on the parabola. Connect points A and B, line segment AB is perpendicular to the x-axis. Point F is the focus of the Parabola. If the distance of AB is 8, find the area of the triangle FAB.

Solution:

What is the parabola look like?

If x = 0, then y = 0. So, the parabola pass through the origin. If y < 0, then y2 > 0, then x > 0. If y > 0, then y2 > 0, then x > 0. So, the parabola is symmetry to the positive x-axis.

the graph of the parabola.

Find the coordinate of the points A and B.

Points A and B lie on the parabola, the segment AB is perpendicular to the x-axis and the length of the segment AB is 8. So, the point A has the coordinate (x, 4) and the point B has the coordinate (x, -4). Because point A lies on the parabola, so point A satisfy the equation of the parabola. The parabola has the equation y2 = 4x and the point A has the coordinate (x, 4). Then substitute the point A into the equation.

y2 = 4x and the point A has the coordinate (x, 4)
42 = 4x
both sides divided by 4
x = 4

So, point A has the coordinate (4, 4) and the point B has the coordinate (4, -4)

Now, we find the coordinate of the focus of the parabola. When the vertex of the parabola is the origin, the graph is symmetry to the x-axis and open right, the coordinate of the focus of the parabola is F (p/2, 0), in which p is related to the equation of the parabola. If a parabola is symmetry to the x-axis and open right, then its equation is y2 = 2px (p > 0) and we are given y2 = 4x. So, 2p = 4, so, p = 2, so the coordinate of the focus of the parabola is F (p/2, 0) = F (1, 0)

find the area of the triangle inside a parabola.

Find the area of the triangle FAB.

Because segment AB is perpendicular to the x-axis and the focus F lies on the x-axis, so, the area of the triangle is:

Area of triangle FAB = (1/2) base × height
= (1/2) × 8 × 3 = 12

note: the distance from the focus F to the segment AB is 4 - 1 = 3.

Therefore, the area of the triangle FAB is 12. For more details, please watch the video.