If the common tangent to the parabolas,
Question:

If the common tangent to the parabolas, $y^{2}=4 x$ and $x^{2}=4 y$ also touches the circle, $x^{2}+y^{2}=c^{2}$, then $c$ is equal to:

  1. (1) $\frac{1}{2 \sqrt{2}}$

  2. (2) $\frac{1}{\sqrt{2}}$

  3. (3) $\frac{}{4}$

  4. (4) $\frac{1}{2}$


Correct Option: , 2

Solution:

 Equation tangent to parabola $y^{2}=4 x$ with slope $m$ be:

$y=m x+\frac{1}{m}$ …(i)

$\because$ Equation of tangent to $x^{2}=4 y$ with slope $m$ be :

$y=m x-a m^{2}$ …….(ii)

From eq. (i) and (ii),

$\frac{1}{m}=-m^{2} \Rightarrow m=-1$

$\therefore$ Equation tangent $: x+y+1=0$

It is tangent to circle $x^{2}+y^{2}=c^{2}$

$\Rightarrow c=\frac{1}{\sqrt{2}}$

 

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