If the line y=mx+7√3 is normal to the hyperbola x^224-y^218=1, then a value of m is:

Question

If the line $y=mx+7\sqrt3$ is normal to the hyperbola $\dfrac{x^2}{24}-\dfrac{y^2}{18}=1$, then a value of $m$ is:

Accepted Answer

Given line y = mx + 7$$\sqrt 3 $$ .....(1)

Given hyperbola

$${{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1$$

Here ${a^2} = 24$ and ${b^2} = 18$

We know the equation of normal to the hyperbola is

$$y = mx \pm {{m\left( {{a^2} + {b^2}} \right)} \over {\sqrt {{a^2} - {b^2}{m^2}} }}$$

$$ \Rightarrow $$ $$y = mx \pm {{m\left( {42} \right)} \over {\sqrt {24 - 18{m^2}} }}$$ .....(2)

Comparing (1) and (2), we get

$${{m\left( {42} \right)} \over {\sqrt {24 - 18{m^2}} }}$$ = $$7\sqrt 3 $$

$$ \Rightarrow $$ 36m2 = 72 - 54m2

$$ \Rightarrow $$ 90m2 = 72

$$ \Rightarrow $$ m2 = $${{72} \over {90}}$$

$$ \Rightarrow $$ m = $${2 \over {\sqrt 5 }}$$

Solutions · 1