Points A , B , C lie on the parabola y 2 = 4 a x . Tangents to the parabola at points A , B , C , when taken in pairs, intersect at points P , Q , R . Determine the ratio ar △ PQR ar △ A BC .

Question

Points A , B , C lie on the parabola y 2 = 4 a x . Tangents to the parabola at points A , B , C , when taken in pairs, intersect at points P , Q , R . Determine the ratio ar △ PQR ar △ A BC ​ .

TestHub · Accepted Answer

To refine the solution, we use the parametric coordinates for a point on the parabola $y^{2} = 4 a x$ as $(a t^{2}, 2 a t)$ . Let the points A, B, C be $(a t_{1}^{2}, 2 a t_{1})$ , $(a t_{2}^{2}, 2 a t_{2})$ , and $(a t_{3}^{2}, 2 a t_{3})$ respectively.

The area of $△ A BC$ is given by:

$ar (△ A BC) = \frac{1}{2} a t_{1}^{2} a t_{2}^{2} a t_{3}^{2} 2 a t_{1} 2 a t_{2} 2 a t_{3} 111$

$= \frac{1}{2} (2 a^{2}) t_{1}^{2} t_{2}^{2} t_{3}^{2} t_{1} t_{2} t_{3} 111 = a^{2} ∣ (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) ∣$

The intersection point of tangents at $t_{1}$ and $t_{2}$ is $P (a t_{1} t_{2}, a (t_{1} + t_{2}))$ . Similarly, $Q (a t_{2} t_{3}, a (t_{2} + t_{3}))$ and $R (a t_{3} t_{1}, a (t_{3} + t_{1}))$ .

The area of $△ PQR$ is given by:

$ar (△ PQR) = \frac{1}{2} a t_{1} t_{2} a t_{2} t_{3} a t_{3} t_{1} a (t_{1} + t_{2}) a (t_{2} + t_{3}) a (t_{3} + t_{1}) 111$

$= \frac{a ^{2}}{2} ∣ (t_{1} - t_{2}) (t_{2} - t_{3}) (t_{3} - t_{1}) ∣$

Therefore, the ratio is:

$\frac{ar ( △ A BC )}{ar ( △ PQR )} = \frac{a ^{2} ∣ ( t _{1} - t _{2} ) ( t _{2} - t _{3} ) ( t _{3} - t _{1} ) ∣}{\frac{a ^{2}}{2} ∣ ( t _{1} - t _{2} ) ( t _{2} - t _{3} ) ( t _{3} - t _{1} ) ∣} = 2$

Mathematics - Parabola Question with Solution | TestHub

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