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10.2 parabola,ellipse,hyperbola — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS10.2 parabola,ellipse,hyperbola1 Mark
MCQ
If the line $y\, = \,mx\, + \,7\sqrt 3 $ is normal to the hyperbola $\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1,$ then a value of $m$ is 
  • ✓
    $\frac{2}{{\sqrt 5 }}$
  • B
    $\frac{{\sqrt 5 }}{2}$
  • C
    $\frac{{\sqrt {15} }}{2}$
  • D
    $\frac{3}{{\sqrt 5 }}$
✓

Answer

Correct option: A.
$\frac{2}{{\sqrt 5 }}$
a
$\frac{{{x^2}}}{{24}} - \frac{{{y^2}}}{{18}} = 1\,\,\,\,\, \Rightarrow a\, = \sqrt {24} :b\, = \sqrt {18} $

Paramentric normal:

$\sqrt {24} \cos \theta .x + \sqrt {18} .y\cot \theta  = 42$

At $x = 0;y = \frac{{42}}{{\sqrt {18} }}\tan \theta  = 7\sqrt 3 $      (from given equation)

$ \Rightarrow \tan \theta  = \sqrt {\frac{3}{2}}  \Rightarrow \sin \theta  =  \pm \sqrt {\frac{3}{5}} $

slope of parametric normal $ = \frac{{ - \sqrt {24} \cos \theta }}{{\sqrt {18} \cot \theta }} = m$

$ \Rightarrow m =  - \sqrt {\frac{4}{3}} \sin \theta  =  - \frac{2}{{\sqrt 5 }}$

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