Question

The equation $\frac{x^2}{2-a}+\frac{y^2}{a-5}+1=0$ represents an ellipse if $a\in$

• A. $(2,\frac{3}{2})\cup (\frac{3}{2},5)$
• B. $(2,\frac{3}{2})$
• C. $(1,\frac{3}{2})$
• D. $(\frac{3}{2},5)$

Answer 1

Answer: (A)

$(2,\frac{3}{2})\cup (\frac{3}{2},5)$

Given $\frac{x^2}{2-a}+\frac{y^2}{a-5}+1=0$

$\frac{x^2}{a-2}+\frac{y^2}{5-a}=1$

For the above equation to be an ellipse

$a-2>0$, $5-a>0$ and $a-2\ne 5-a$

$a>2$, $a<5$ and $a\ne \frac{7}{2}$

$aϵ(2,\frac{7}{2})\bigcup (\frac{7}{2},5)$