The values of $m$ for which the line $y=m x+2$ becomes a tangent to the hyperbola $4 x^2-9 y^2=36$ is

The values of $m$ for which the line $y=m x+2$ becomes a tangent to the hyperbola $4 x^2-9 y^2=36$ is
  1. $\pm \frac{2}{3}$
  2. $\pm \frac{2 \sqrt{2}}{3}$
  3. $\pm \frac{8}{9}$
  4. $\pm \frac{4 \sqrt{2}}{3}$

Solution

We have, line, $y=m x+2$... (i) Hyperbola, $4 x^2-9 y^2=36... (ii)$ On solving eqs. (i) and (ii), we get $x^2\left(4-9 m^2\right)-36 m x-72=0$ Since, this line is a tangent of hyperbola $\begin{array}{ll} \therefore & \mathrm{D}=\mathrm{b}^2-4 \mathrm{ac}=0 \\ \therefore & (36 m)^2+4 \times 72\left(4-9 m^2\right)=0 \\ \Rightarrow & 36 \times 36 m^2+4 \times 36 \times 2\left(4-9 m^2\right)=0 \\ \Rightarrow & 9 m^2+8-18 m^2=0 \\ \Rightarrow & 9 m^2=8 \\ \Rightarrow & m^2=\frac{8}{9} \\ \therefore & m^2= \pm \sqrt{\frac{8}{9}}= \pm \frac{2 \sqrt{2}}{3} \end{array}$

Asked in: AP EAMCET 2016

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