Question

The equations of the tangents drawn at the ends of the major axis of the ellipse $9x^2+5y^2-30y=0$ are

• A. $y=\pm 3$
• B. $x=\pm \sqrt{5}$
• C. $y=0,y=6$
• D. None of the above

Answer 1

Answer: (C)

Given equation of ellipse is
$9x^2+5y^2-30y=0$
The above equation can be rewritten as
$9x^2+5\left(y^2-6y\right)=0$
$\Rightarrow 9x^2+5(y-6y+9)=45$
$\Rightarrow 9x^2+5(y-3{)}^2=45$
$\Rightarrow \frac{x^2}{5}+\frac{(y-3{)}^2}{9}=1$
Since, $a^2<b^2$, so axis of ellipse is $Y$ -axis.
To find vertex at $Y$ -axis, put $x=0$.
$\therefore 5(y-3{)}^2=45$
$\Rightarrow (y-3{)}^2=9$
$\Rightarrow y-3=\pm 3$
$\Rightarrow y=0$ or $y=6$
Therefore, tangents at vertex are $y=0,y=6$