# Two common tangents to the circle $\dpi{100} x^{2}+y^{2}=2a^{2}$  and parabola $\dpi{100} y^{2}=8ax\; are$ Option 1) $x=\pm (y+2a)$ Option 2) $y=\pm (x+2a)$ Option 3) $x=\pm (y+a)$ Option 4) $y=\pm (x+a)$

S Sabhrant Ambastha

As we learnt in

Equation of tangent -

$y= mx+\frac{a}{m}$

- wherein

Tengent to $y^{2}=4ax$ is slope form.

and

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 circle x2+y2=2a2

$is \: y=mx+\sqrt{2}a\sqrt{\left 1+m^{2} \right }$

and Tangent to parabola y2=8ax

$is \: y=mx+\frac{2a}{m}$

$we \: get \: \sqrt{2a} \: \sqrt{1+m^{2}}= \frac{2a}{m}$

$\\ (m^{2}+1)\cdot m^{2}=2 \\ \\ m^{4}+2m^{2}-m^{2}-2=0 \\ \\ m^{2}=1 \: \: \ ; \: \: \: m= \pm 1 \\ \\ we \: get \: \: \: y=\pm (x+2a)$

Option 1)

$x=\pm (y+2a)$

This option is incorrect

Option 2)

$y=\pm (x+2a)$

This option is correct

Option 3)

$x=\pm (y+a)$

This option is incorrect

Option 4)

$y=\pm (x+a)$

This option is incorrect

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