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A straight line touches both \mathrm{ x^2+y^2=8 } and  \mathrm{ y^2=16 x } . Its equation is  \mathrm{ y= \pm(x+k)} , where  k=

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

The equation of any tangent to the circle \mathrm{ x^2+y^2=8} is \mathrm{ x \cos \theta+y \sin \theta=2 \sqrt{ } 2}              \mathrm{.....(1)}

The equation of any tangent to the parabola \mathrm{y^2=16 x} is

\mathrm{y=m x+\frac{4}{m}}                                                                               \mathrm{.....(2)}

Since (1) and (2) are identical

\mathrm{ \frac{\cos \theta}{-m}=\frac{\sin \theta}{1}=\frac{m}{\sqrt{2}} \Rightarrow \cos \theta=\frac{-m^2}{\sqrt{2}} \text { and } \sin \theta=\frac{m}{\sqrt{2}} }

Squaring and adding, \mathrm{m^4+m^2-2=0}

\mathrm{ \Rightarrow \mathrm{m}^2=1 \Rightarrow \mathrm{m}= \pm 1 }

Substituting in (2) the equation of the required tangent is \mathrm{y= \pm(x+4)}

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