Question

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

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is C $2$

Equation of tangent to ellipse:- $y=mx+\sqrt{a^2{m}^2+b^2}$

Here, $a^2=\frac{32}{3},b^2=\frac{32}{5}$

$\Rightarrow 5=3m+\sqrt{\frac{32m^2}{3}+\frac{32}{5}}$

$\Rightarrow 5-3m=4\sqrt{\frac{2m^2}{3}+\frac{2}{5}}$

$\Rightarrow 25+9m^2-30m=16\left(\frac{10m^2+6}{15}\right)$

$\Rightarrow m^2(160-135)+450m+96-375=0$

$\Rightarrow 25m^2+450m-279=0$

$D=(450{)}^2+279(100)=100(2025+279)>0$

$\Rightarrow$ Two solutions

$\Rightarrow \left(C\right)$.