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If the common tangents to the parabola, x²=4y and the circle, x²+y²=4 intersect at the point P, then the distance of P from the origin, is :

Option 1)
$\sqrt{2} + 1$

Option 2)
$2\left ( 3+2\sqrt{2} \right )$

Option 3)
$2 \left ( \sqrt{2} +1\right )$

Option 4)
$3 + 2\sqrt{2}$

Answers (2)

As we learnt in

Standard equation of parabola -

$x^{2}=4ay$

- wherein

Condition of tangency -

$c^{2}=a^{2}\; (1+m^{2})$

- wherein

If $y=mx+c$ is a tangent to the circle $x^{2}+y^{2}=a^{2}$

Tangent to $x^{2}+y^{2}=4$ is

$y=mx\pm 2\sqrt{1+m^{2}}$

Also $x^{2}=4y$

$x^{2}=4mx+8\sqrt{1+m^{2}}$

If we put D=0

$m^{4}-4m^{2}-4=0$

$m^{2}=2+2\sqrt{2}$

$m^{2}=2(\sqrt{2}+1)$

Option 1)

$\sqrt{2} + 1$

This option is incorrect

Option 2)

$2\left ( 3+2\sqrt{2} \right )$

This option is incorrect

Option 3)

$2 \left ( \sqrt{2} +1\right )$

This option is correct

Option 4)

$3 + 2\sqrt{2}$

This option is incorrect

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Vakul

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