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  4. The focal chord of the parabola (y-2)2=16(x-1) is a tangent to the circle x2+y2-14 x-4 y+51=0, then slope of the focal chord can be

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Q. The focal chord of the parabola $(y-2)^{2}=16(x-1)$ is a tangent to the circle $x^{2}+y^{2}-14 x-4 y+51=0, $ then slope of the focal chord can be

Conic Sections

Solution:

Focus of given parabola is $(5,2)$.
Now, any line through $(5,2)$ is $(y-2)=m(x-5)$
This will be a tangent to the circle $(x-7)^{2}+(y-2)^{2}=2$,
if $\left|\frac{0-2 m}{\sqrt{1+ m ^{2}}}\right|=\sqrt{2} \Rightarrow 4 m ^{2}=2+2 m ^{2} \Rightarrow m = \pm 1 .$