CUET PG 2026 — Mathematics PYQ

Solution

Step 1: Find the foci of the given ellipse

The equation of the ellipse is:

Comparing with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$:

$a^2=25⟹a=5$

$b^2=9⟹b=3$

The eccentricity $e_e$ of the ellipse is:

The foci of the ellipse are $(\pm a{e}_e,0)$:

Step 2: Use the hyperbola's properties

The foci of the hyperbola are the same as the ellipse, so for the hyperbola:

Given that the eccentricity of the hyperbola $e_h=2$:

Step 3: Find $B^2$ for the hyperbola

For a hyperbola, the relation between $A$, $B$, and $e$ is:

Substituting the values:

Step 4: Write the equation of the hyperbola

The standard equation of the hyperbola is:

Substituting $A^2=4$ and $B^2=12$:

Conclusion

The equation of the hyperbola is $\frac{x^2}{4}-\frac{y^2}{12}=1$.

The correct option is (a).