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If two tangents are drawn from point \mathrm{P\left(x_1, y_1\right)} to \mathrm{y^2=4 a x} and the length of the chord of contact is

\mathrm{ \frac{k}{a} \sqrt{\left(y_1^2-4 a x_1\right)\left(y_1^2+4 a^2\right)} }

where k is equal to

Option: 1

1


Option: 2

\frac{1}{2}


Option: 3

\frac{1}{4}


Option: 4

2


Answers (1)

best_answer

The equation of QR is

                   \mathrm{ T=0 }
Or  \mathrm{y y_1-2 a\left(x+x_1\right)=0}               ..........(1)       

The parabola is \mathrm{y^2-4 a x=0}             ..........(2)

\mathrm{\therefore \quad}  the y-coordinates of points of intersection Q and R are given by [use (1) and (2) to eliminate x ]

\mathrm{ y^2=4 a\left(\frac{y y_1-2 a x_1}{2 a}\right) }
Or \mathrm{y^2-2 y y_1+4 a x_1=0}              ..........(3)

Let Q and R be \mathrm{\left(h_1, k_1\right),\left(h_2, k_2\right)} respectively.

\mathrm{ \therefore \quad k_1+k_2=2 y_1 \text { and } k_1 k_2=4 a x_1 }       ..........(4)

As Q and R lie on parabola, the conditions are

\mathrm{ \begin{aligned} & k_1^2=4 a h_1 \text { and } k_2^2=4 a h_2\, \, \, \, \, .....(5) \\\\ \therefore & k_1^2-k_2^2=4 a\left(h_1-h_2\right) \\\\ \Rightarrow & \frac{k_2^2-k_1^2}{4 a}=h_2-h_1 \end{aligned} }..........(6)

Hence, \mathrm{Q R^2=\left(k_2-k_1\right)^2+\left(h_2-h_1\right)^2}
 \mathrm{ \begin{aligned} & =\left(k_2-k_1\right)^2+\left(\frac{k_2^2-k_1^2}{4 a}\right)^2 \\\\ & =\left(k_2-k_1\right)^2\left[1+\frac{\left(k_2+k_1\right)^2}{16 a^2}\right] \end{aligned} }

\mathrm{ \begin{aligned} & =\left[\left(k_2+k_1\right)^2-4 k_1 k_2\right]\left[1+\frac{\left(k_2+k_1\right)^2}{16 a^2}\right] \\\\ & =\left[4 y_1^2-4\left(4 a x_1\right)\right]\left[1+\frac{\left(2 y_1\right)^2}{16 a^2}\right] \, \, \, \, \, \, \, by\, \, using (4)\\\\ & =4\left(y_1^2-4 a x_1\right)\left(\frac{4 a^2+y_1^2}{4 a^2}\right) \\\\ & =\frac{1}{a^2}\left(y_1^2-4 a x_1\right)\left(y_1^2+4 a^2\right) \\\\ & \therefore \quad Q R=\frac{1}{a} \sqrt{\left(y_1^2-4 a x_1\right)\left(y_1^2+4 a^2\right)} \\\\ & \Rightarrow \quad k=1 \\ & \end{aligned} }
 

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