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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

Posted by

Vakul

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