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10-2. Parabola, Ellipse, Hyperbola
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For the Hyperbola ${x^2}{\sec ^2}\theta - {y^2}cose{c^2}\theta = 1$ which of the following remains constant when $\theta $ varies $= ?$
A
Focus
B
directrix
C
eccentricity
D
lenght of Latus rectum
(AIEEE-2007)
Solution
Given equation is comparing on $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
We get
$a^{2}=\cos ^{2} a$ and $b^{2}=\sin ^{2} a$
$\therefore \sin ^{2} a+\cos ^{2} a=a^{2}+b^{2}$
$\Rightarrow e-\frac{\overline{a^{2}+b^{2}}}{a^{2}}$
Now,
$=\frac{\overline{1}}{\cos ^{2} a}=\frac{1}{\cos a}$
Now, foci $a e=\cos a \cdot \frac{1}{\cos \alpha}=1$
Standard 11
Mathematics
