Question

The equation $\frac{x^2}{12-k}+\frac{y^2}{8-k}=1$ represents.

• A. A hyperbola, if $k<8$
• B. A ellipse, if $k>8$
• C. A hyperbola, if $8<k<12$
• D. None of the above

Answer 1

The correct option is C

A hyperbola, if $8<k<12$

You try to imagine first the standard forms of hyperbola and ellipse.

Hyperbola $\to$ $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ or $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$

Ellipse $\to$ $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

This shows that for the curve to become a hyperbola the

coefficients should be of opposite signs.

Applying this condition for the given curve.

$(12-k)(8-k)<0$

i.e., $(12-k)(k-8)<0$

If the given curve is an ellipse the coefficients of $x^2$ and $y^2$ should

be positive sign.For the curve to be an ellipse.

i.e.,$12-k<0and8-k<0$

i.e.,$k<12andk<0$

Hence option (c)