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  • The ratio of area of triangle formed by the pair of tangents at the extremities of focal chord and area of triangle formed by the same focal chord at the vertex of parabola <img alt="\mathrm{y^2=4

The ratio of area of triangle formed by the pair of tangents at the extremities of focal chord and area of triangle formed by the same focal chord at the vertex of parabola \mathrm{y^2=4 x}\mathrm{y^2=4 x} is how much more than arithmetic mean of the abscissa of focal chord.

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

\frac{1}{2}


Answers (1)

best_answer

Area of triangle  \mathrm{A B C, t_1 t_2=-1}

\mathrm{\begin{aligned} & \quad\left|\begin{array}{ccc} -1 & t_1+t_2 & 1 \\ t_1^2 & 2 t_1 & 1 \\ t_2^2 & 2 t_2 & 1 \end{array}\right|=\left(t_1-t_1\right) \\\\ & =\left(t_2^2+t_1^2+2\right) \end{aligned}}

\mathrm{\begin{aligned} & \text { and area of triangle } \mathrm{ABO}= \\ & \left|\begin{array}{ccc} 0 & 0 & 1 \\ t_1^2 & 2 t_1 & 1 \\ t_2^2 & 2 t_2 & 1 \end{array}\right| \\ & =2\left(\mathrm{t}_2-\mathrm{t}_1\right) \\ & \Rightarrow \frac{\text { Area }(\triangle \mathrm{ABC})}{\text { Area }(\triangle \mathrm{ABO})}=\left(\frac{\mathrm{t}_1^2+\mathrm{t}_2^2}{2}+1\right) \text {. } \\ & \end{aligned}}

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