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Maths-

General

Easy

Question

Tangents are drawn from the point (-1,2) on the parabola $y to the power of 2 end exponent equals 4 x$ The length, these tangents will intercept on the line $x equals 2$ :

6
6 $\sqrt{2}$
2 $\sqrt{6}$
none of these

hint Hint:

The correct answer is: 6 $square root of 2$

The length of tangent is asked to be find
if the line $x = 2$ intersects these tangents at $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ then the length of the intercept is given by $| y_{1} - y_{2} |$
$S S_{1} = {T2}$ is the equation of pair of tangents
$({y2} - 4 x) (8) = 4 (y - x + 1)^{2}$
$\Rightarrow {y2} - 2 y (1 - x) - ({x2} + 6 x + 1) = 0$
Put $x = 2$
$\Rightarrow {y2} + 2 y - 17 = 0$
Clearly, $y_{1}, y_{2}$ are the roots of the equation ${y2} + 2 y - 17 = 0$
$\Rightarrow y_{1} + y_{2} = - 2$
$y_{1} \cdot y_{2} = - 17$
Now, $| y_{1} - y_{2} | = \sqrt{(y_{1} + y_{2} {)2} - 4 y_{1} \cdot {y2}}$
$= \sqrt{2^{2} - 4 (- 17)}$
$= \sqrt{72}$
$= 6 \sqrt{2}$
$∴ | y_{1} - y_{2} | = 6 \sqrt{2}$

Hence finally the length is $6 square root of 2$

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