If the line y=m x+7


If the line $y=m x+7 \sqrt{3}$ is normal to the hyperbola $\frac{x^{2}}{24}-\frac{y^{2}}{18}=1$, then a value of $\mathrm{m}$ is :

  1. (1) $\frac{\sqrt{5}}{2}$

  2. (2) $\frac{\sqrt{15}}{2}$

  3. (3) $\frac{2}{\sqrt{5}}$

  4. (4) $\frac{3}{\sqrt{5}}$

Correct Option: , 3


Since, $1 \mathrm{x}+\mathrm{my}+\mathrm{n}=0$ is a normal to $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$,

then $\frac{a^{2}}{l^{2}}-\frac{b^{2}}{m^{2}}=\frac{\left(a^{2}+b^{2}\right)^{2}}{n^{2}}$

but it is given that $m x-y+7 \sqrt{3}$ is normal to hyperbola


then $\frac{24}{m^{2}}-\frac{18}{(-1)^{2}}=\frac{(24+18)^{2}}{(7 \sqrt{3})^{2}} \Rightarrow m=\frac{2}{\sqrt{5}}$

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