KVPY Sample Paper KVPY Stream-SX Model Paper-3

Equation of a common tangent to the circle, \[{{x}^{2}}+{{y}^{2}}-6x=0\] and the parabola, \[{{y}^{2}}=4x,\] is:

A) \[2\sqrt{3}y=12x+1\]

B) \[2\sqrt{3}y=-x-12\]

C) \[\sqrt{3}y=x+3\]

D) \[\sqrt{3y}=3x+1\]

Correct Answer: C

Solution :

Tangent to \[{{y}^{2}}=\,4x\] with slope m is \[y\,=\,mx\,+\frac{1}{m}\]\[{{m}^{2}}x-my+1=0\] is tangent to the circle \[{{x}^{2}}+{{y}^{2}}-6x=0\]

\[\Rightarrow \] \[d=r\]

\[\therefore \] \[\left| \frac{3{{m}^{2}}+1}{\sqrt{{{m}^{4}}+{{m}^{2}}}} \right|\,=\,3\]

\[\Rightarrow \] \[\text{m}\,=\,\pm \,\frac{1}{\sqrt{3}}\]

For \[m\,=\,\frac{1}{\sqrt{3}},\]is tangent to the circle \[y\,=\,\frac{x}{\sqrt{3}}+\sqrt{3}\]

i.e., \[\sqrt{3y}\,=\,x+3.\]