Question

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

• A. 0
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

Equation of ellipse are
$S_1:\frac{x^2}{32/3}+\frac{y^2}{32/5}=1$ and $\frac{x^2}{450/25}+\frac{y^2}{450/9}=1$
$\Rightarrow S_2:\frac{x^2}{18}+\frac{y^2}{50}=1$
Putting $(3,5)$ in the equation of ellipse, we get $S-1>0$ and $S_2=0$ so that $(3,5)$ lies outside $S_1$ and hence two tangents can be drawn through the point $(3,5).$ The point $(3,5)$ lies on $S_2=0$ and only one tangent can be drawn.
Thus, the total number of tangents passing through $(3,5)$ to the ellipse is $2+1=3$