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  • Let the eccentricity of the hyperbola <img alt="\mathrm{H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1\; be\; \sqrt{\frac{5}{2}}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7BH%3

Let the eccentricity of the hyperbola \mathrm{H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1\; be\; \sqrt{\frac{5}{2}}} and length of its latus rectum be \mathrm{6\sqrt{2},\: If\:y= 2x+c } is a tangent to the hyperbola \mathrm{H }, then the value of \mathrm{c^{2} } is equal to

Option: 1

18


Option: 2

20


Option: 3

24


Option: 4

32


Answers (1)

best_answer

\mathrm{e^{2}=1+\frac{b^{2}}{a^{2}}=\frac{5}{2}\: \Rightarrow \frac{b^{2}}{a^{2}}= \frac{3}{2} \quad --(1)}
\mathrm{\frac{2b^{2}}{a^{2}}=6\sqrt{2}\; \Rightarrow \; 2b^{2}= 6\sqrt{2}\, a\: \Rightarrow 3a^{2}= 6\sqrt{2}\, a\Rightarrow a= 2\sqrt{2},b= 2\sqrt{3}.}

Equation of tangent of slope  \mathrm{m=2 } is

\mathrm{y= 2x+\sqrt{a^{2}\times 2^{2}-b^{2}}}
\mathrm{y= 2x+\sqrt{32-12} \quad c=\sqrt{20} \; \; \Rightarrow c^{2}= 20}

Option (B)
 

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Riya

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