Question

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

• A. $4c^2=369$
• B. $c^2=369$
• C. $8m+5=0$
• D. $5m=4$

Answer 1

The correct option is A

$4c^2=369$

Explanation of the correct option:

Option $\left(A\right)$: $4c^2=369$

Given: The line $y=mx+c$ is a common tangent to the hyperbola $\left(\frac{x^2}{100}\right)-\left(\frac{y^2}{64}\right)=1$ and the circle $x^2+y^2=36$.

We know that for hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ , $c$ is given by, $c=\sqrt{a^2{m}^2-b^2}$

$\Rightarrow c=\sqrt{100m^2-64}$…………………$\left(1\right)$

For he circle $x^2+y^2=a^2$, $c$ is given by, $c=a\sqrt{1+m^2}$

$\Rightarrow c=6\sqrt{1+m^2}$…………………$\left(2\right)$

Since the line $y=mx+c$ is a common tangent to the hyperbola and the circle,

Thus, $\sqrt{100m^2-64}=6\sqrt{1+m^2}$

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

$\Rightarrow$ $100m^2-64=36+36m^2$

$\Rightarrow$ $64m^2=100$

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

$\Rightarrow$ $m=\pm \frac{10}{8}$

From equation $\left(2\right)$,

$$\begin{gathered} c=6\sqrt{1+\frac{100}{64}} \\ \Rightarrow c=6\left(\frac{1}{8}\sqrt{164}\right) \end{gathered}$$

Taking square both side,

$$\begin{gathered} \Rightarrow c^2=\frac{36}{64}\left(164\right) \\ \Rightarrow c^2=\frac{369}{4} \\ \Rightarrow 4c^2=369 \end{gathered}$$

Hence option $\left(A\right)$ is correct option.

Explanation of the incorrect option.

Option $\left(B\right)$: $c^2=369$

Since, $4c^2=369$

$\Rightarrow c^2=92.25$

Hence option $\left(B\right)$ is incorrect option.

Option $\left(C\right)$: $8m+5=0$

Since, $m=\pm \frac{10}{8}$

For $m=\frac{10}{8}$

$\begin{array}{rcl}8m+5& =& 8\left(\frac{10}{8}\right)+5\\ & =& 15\end{array}$

For $m=-\frac{10}{8}$

$\begin{array}{rcl}8m+5& =& 8\left(-\frac{10}{8}\right)+5\\ & =& -5\end{array}$

Hence option $\left(C\right)$ is incorrect option.

Option $\left(D\right)$: $5m=4$

Since, $m=\pm \frac{10}{8}$

For $m=\frac{10}{8}$

$\begin{array}{rcl}5m& =& 5\times \frac{10}{8}\\ & =& 6.25\end{array}$

For $m=-\frac{10}{8}$

$\begin{array}{rcl}5m& =& 5\times \left(-\frac{10}{8}\right)\\ & =& -6.25\end{array}$

Hence option $\left(D\right)$ is incorrect option.

Hence option $\left(A\right)$ is the correct option.