Question

The equation $\frac{x^2}{10-\lambda }+\frac{y^2}{6-\lambda }=1$ represents

• A. a hyperbola if $\lambda <6$
• B. an ellipse if $\lambda >6$
• C. a hyperbola if $6<\lambda <10$
• D. an ellipse if $0<\lambda <6$

Answer 1

Answer: (C), (D)

a hyperbola if $6<\lambda <10$
an ellipse if $0<\lambda <6$

The general equation of an ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and that of a hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Using this, we get that the above equation is an ellipse if $10-\lambda >0$ and $6-\lambda >0$. The combined solution gives $\lambda <6.$
For a hyperbola, the coefficient of $x^2$ and $y^2$ must be of opposite sign. Hence,
$10-\lambda >0$ and $6-\lambda <0$ which gives $6<\lambda <10$