Question

For hyperbola $\frac{x^2}{co{s}^{2}a}-\frac{y^2}{si{n}^{2}a}=1$ which of the following remains constant with change in 'a'?

• A. Abscissae of vertices
• B. Abscissae of foci
• C. Eccentricity
• D. Directrix

Answer 1

Answer: (B)

Abscissae of foci

Comparing given equation $\frac{x^2}{co{s}^{2}a}-\frac{y^2}{si{n}^{2}a}=1$ with the standard equation $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, we get
$a^2=co{s}^{2}a$ and $b^2=si{n}^{2}a$
$\therefore 1=si{n}^{2}a+co{s}^{2}a=a^2+b^2$
$\Rightarrow e=\sqrt{\frac{a^2+b^2}{a^2}}$
$\Rightarrow e=\frac{1}{cosa}$
Now, foci $ae=cosa.\frac{1}{cosa}=1$