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


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. $5 \mathrm{~m}=4$

  2. $4 \mathrm{c}^{2}=369$

  3. $c^{2}=369$

  4. $8 m+5=0$

Correct Option: , 2


$\mathrm{y}=\mathrm{m} \mathrm{x}+\mathrm{c}$ is tangent to

$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and $x^{2}+y^{2}=36$

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

$\Rightarrow 100 \mathrm{~m}^{2}-64=36+36 \mathrm{~m}^{2}$

$\mathrm{m}^{2}=\frac{100}{64} \Rightarrow \mathrm{m}=\pm \frac{10}{8}$

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

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

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