If the line y=m x+c is a common tangent to

Question:

If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36$, then which one of the following is true?

  1. (1) $c^{2}=369$

  2. (2) $5 m=4$

  3. (3) $4 c^{2}=369$

  4. (4) $8 m+5=0$


Correct Option: , 3

Solution:

General tangent to hyperbola in slope form is

$y=m x \pm \sqrt{100 m^{2}-64}$

and the general tangent to the circle in slope form is

$y=m x \pm 6 \sqrt{1+m^{2}}$

For common tangent,

$36\left(1+m^{2}\right)=100 m^{2}-64$

$\Rightarrow 100=64 m^{2} \Rightarrow m^{2}=\frac{100}{64}$

$\therefore c^{2}=36\left(1+\frac{100}{64}\right)=\frac{164 \times 36}{64}=\frac{369}{4}$

$\Rightarrow 4 c^{2}=369$

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