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  4. The equation of the tangent to the parabola y2=4x whose slope is positive and which also touches x2+y2=(1/2) is

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Q. The equation of the tangent to the parabola $y^{2}=4x$ whose slope is positive and which also touches $x^{2}+y^{2}=\frac{1}{2}$ is

NTA AbhyasNTA Abhyas 2022

Solution:

Tangent to parabola is $y=mx+\frac{1}{m}$
Tangent to circle is $y=mx\pm\frac{1}{\sqrt{2}}\sqrt{1 + m^{2}}$
For common tangent
$\frac{1}{m^{2}}=\frac{1}{2}\left(1 + m^{2}\right)\Rightarrow m^{4}+m^{2}-2=0$
$\Rightarrow m^{2}=1\Rightarrow m=1$