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10-2. Parabola, Ellipse, Hyperbola
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અતિવલય $\frac{{{x^2}}}{{{{\cos }^2}\alpha }}\,\, - \,\,\frac{{{y^2}}}{{{{\sin }^2}\,\,\alpha }}\, = \,\,1\,$ માટે જ્યારે $\,\alpha $ બદલાતો હોય ત્યારે નીચેના માંથી કયું પદ અચળ રહે.
A
શિરોબિંદુઓ
B
નાભિઓ
C
ઉત્કેન્દ્રતા
D
નિયામિકા
Solution
Given equation of hyperbola is $\frac{ x ^2}{\cos ^2 \alpha}-\frac{ y ^2}{\sin ^2 \alpha}=1$
Here, $a^2=\cos ^2 \alpha$ and $b^2=\sin ^2 \alpha$
(i.e, comparing with standard equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ )
We know, foci $=(\pm a ae , 0)$
where $a e=\sqrt{a^2+b^2}=\sqrt{\cos ^2 \alpha+\sin ^2 \alpha}=1$
$\Rightarrow \text { foci }=(\pm 1,0)$
where as vertices are $(\pm \cos \alpha, 0)$
Eccentricity ae $=1 \Rightarrow e =\frac{1}{\cos \alpha}$
Hence foci reamins constant with change in $\alpha$ '
Standard 11
Mathematics
