Question

The line $3x+5y=k$ is a tangent to the ellipse $16x^2+25y^2=400,$ if $k$ is:

• A. $\pm 5$
• B. $\pm 15$
• C. $\pm 25$
• D. $\pm 10$

Answer 1

The correct option is C $\pm 25$

Given line and the ellipse may be written as $y=-\frac{3}{5}x+\frac{k}{5}$ and $\frac{x^2}{25}+\frac{y^2}{16}=1$

$\Rightarrow a^2=25,b^2=16,c=\frac{k}{5},m=-\frac{3}{5}$

Now using condition of tangency, $c^2=a^2{m}^2+b^2$

$\Rightarrow \frac{k^2}{5^2}=25\times \frac{3^2}{5^2}+16=25$

$\therefore k=\pm 25$