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Points A, B and C lie on the parabola \mathrm{y^2=4 a x} The tangents at A, B and C taken in pairs, intersect at the points P, Q and R. Determine the ratio of the areas of the triangles ABC and PQR.

 

 

 

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

\mathrm{1 / 2}


Answers (1)

best_answer

Let \mathrm{A\left(a t_1^2, 2 a t_1\right), B\left(a t_2^2, 2 a t_2\right), C\left(a t_3^2, 2 a t_3\right)}  be the three points on the parabola and let the tangents at A, B and C meet to form the triangle PQR.

\mathrm{\text { Then } P\left[a t_1 t_2, a\left(t_1+t_2\right)\right], Q\left[a t_3 t_1, a\left(t_3+t_1\right)\right], R\left[a t_2 t_3, a\left(t_2+t_3\right)\right]}

Then area of ΔABC =\mathrm{\frac{1}{2}\left|\begin{array}{lll} a t_1^2 & 2 a t_1 & 1 \\ a t_2^2 & 2 a t_2 & 1 \\ a t_3^2 & 2 a t_3 & 1 \end{array}\right|=a^2\left|\left(t_1-t_2\right)\left(t_2-t_3\right)\left(t_3-t_1\right)\right|}

(taking the magnitude)

\mathrm{\text { Area of } \triangle \mathrm{PQR}=\frac{1}{2}\left|\begin{array}{lll} a t_1 t_2 & a\left(t_1+t_2\right) & 1 \\ a t_3 t_1 & a\left(t_3+t_1\right) & 1 \\ a t_2 t_3 & a\left(t_2+t_3\right) & 1 \end{array}\right|=\frac{a^2}{2}\left|\left(t_1-t_2\right)\left(t_2-t_3\right)\left(t_3-t_1\right)\right|}

\mathrm{\text { Hence } \frac{\text { Area of } \triangle \mathrm{ABC}}{\text { Area of } \triangle \mathrm{PQR}}=2}

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