Question

The length of the transverse axis of a hyperbola is 2cos t. The foci of the hyperbola are the same as that of the ellipse $9x^2+16y^2=144$.
The equation of the hyperbola is ___

• A. $\frac{x^2}{co{s}^{2}t}-\frac{y^2}{7-co{s}^{2}t}=1$
• B. $\frac{x^2}{co{s}^{2}t}-\frac{y^2}{7+co{s}^{2}t}=1$
• C. $\frac{x^2}{1+co{s}^{2}t}-\frac{y^2}{7-co{s}^{2}t}=1$
• D. $\frac{x^2}{1+co{s}^{2}t}-\frac{y^2}{7+co{s}^{2}t}=1$

Answer 1

The correct option is A

$\frac{x^2}{co{s}^{2}t}-\frac{y^2}{7-co{s}^{2}t}=1$

For the given hyperbola, Length of the transverse axis = 2a
$\text{2a= 2cos t}\Rightarrow \text{a = cos t}$

Consider the ellipse

${\text{9x}}^2+{\text{16y}}^2=144\Rightarrow \frac{{\text{x}}^2}{16}+\frac{{\text{y}}^2}{9}=1\text{Focal length, c=}\sqrt{16-9}=\sqrt{7}$

This is the same as the focal length of the hyperbola.

Now, for the hyperbola,
${\text{b}}^2={\text{c}}^2-{\text{a}}^2{\text{b}}^2=7-{\text{cos}}^2\text{t}$

The equation of the hyperbola is
$\frac{{\text{x}}^2}{{\text{cos}}^2\text{t}}-\frac{{\text{y}}^2}{7-{\text{cos}}^2\text{t}}=1$