Question

The total number of tangents through the point $(3,5)$ that can be drawn to the ellipses $3x^2+5y^2=32$ and $25x^2+9y^2=450$ is

• A. $0$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$3$

Let $S_1=3x^2+5y^2-32$
and $S_2=25x^2+9y^2-450$
At point $(3,5)$
$S_1=3{\left(3\right)}^2+5{\left(5\right)}^2-32=120>0$
and $S_2=25{\left(3\right)}^2+9{\left(5\right)}^2-450$
$=225+225-450$
$=0$
$\therefore$ Point $(3,5)$ lies outside the first ellipse and for second ellipse lies on the ellipse.
Hence, two tangents for the first ellipse and one tangent for second ellipse can be drawn.