Question

If the ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5b+7$, then $b$ does not lie in

• A. $[4,5]$
• B. $(\infty ,2)\cup (3,\infty )$
• C. $(-\infty ,0)$
• D. $[2,3]$

Answer 1

The correct option is D $[2,3]$

For the two ellipses to intersect at four distinct points, $a>1$

$\Rightarrow$ $b^2-5b+7>1$

$\Rightarrow$ $b^2-5b+6>0$

$\Rightarrow$ $b\in (-\infty ,2)\cup (3,\infty )$

$\Rightarrow b$ does not lie in $[2,3]$