Question

find the area of the triangle formed by the lines joining the focus of the parabola y square = 4x to the points on it which have abscissa equal to 16

Answer 1

Please see the attachment

Answer 2

The area is 120 sq. units.

Step-by-step explanation:

The equation of the parabola is y² = 4x ........ (1)

Now, the focus of a parabola having equation y² =4ax is given by the point (a,0).

Now, in our case a = 1, so, the focus will be at (1,0).

Let the point on the parabola whose abscissa equal to 16 is (16,k).

So, it will satisfy the equation (1) and k² = 16 × 4

⇒ k = ± 8

Therefore, the points are (16,8) and (16,-8).

Now, the area of the triangle joining those two points on the parabola and the focus (1,0) will be given by

Δ = $\frac{1}{2}|1(8 - (- 8)) + 16(- 8 - 0) + 16(0 - 8)|$

⇒ Δ = 120 sq, units. (Answer)

We know the area of a triangle with vertices are  $(x_{1},y_{1})$, $(x_{2},y_{2})$ and $(x_{3},y_{3})$, given by the formula

Δ = $\frac{1}{2} |x_{1}(y_{2} - y_{3} ) + x_{2}(y_{3} - y_{1} ) + x_{3}(y_{1} - y_{2} ) |$