Which one of the following is independent of α in the hyperbola (0 lt α lt π / 2) (x^2)/(cos^2 α)-(y^2)/(sin^2 α)=1

Correct answer: (B) Abscissa of foci. e^2=1+(b^2)/(a^2)=1+(sin^2 α)/(cos^2 α)=1/(cos^2 α) and a^2=cos^2 α ∴ a^2 e^2=1 ∴ foci ( ± a e, 0)=( ± 1,0) , which is independent of α .

Q100 / 200 · Manipal (MET) 2013 · Mathematics

Which one of the following is independent of $\alpha$ in the hyperbola $\quad(0 \lt \alpha \lt \pi / 2)$ $\frac{x^2}{\cos ^2 \alpha}-\frac{y^2}{\sin ^2 \alpha}=1$

A
Eccentricity
Correct answer
B
Abscissa of foci
C
Directrix
D
Vertex

Solution

$e^2=1+\frac{b^2}{a^2}=1+\frac{\sin ^2 \alpha}{\cos ^2 \alpha}=\frac{1}{\cos ^2 \alpha}$ and $\quad a^2=\cos ^2 \alpha$ $\therefore \quad a^2 e^2=1$ $\therefore$ foci $( \pm a e, 0)=( \pm 1,0)$, which is independent of $\alpha$.

Hence, the correct answer is option: 'B'.

Which one of the following is independent of $\alpha$ in the hyperbola $\quad(0 \lt \alpha \lt \pi / 2)$ $\frac{x^2}{\cos ^2 \alpha}-\frac{y^2}{\sin ^2 \alpha}=1$

Solution

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