Question

If $P(\alpha ,\beta )$ is a point on the ellipse $4x^2+9y^2=1$, at which the tangent is parallel to the line $y=\frac{8x}{9}+200,$ then the value of $|\alpha +\beta |$ is

• A. $\frac{1}{5}$
• B. $5$
• C. $\frac{3}{5}$
• D. $\frac{5}{3}$

Answer 1

The correct option is A $\frac{1}{5}$

Equation of tangent to the ellipse $4x^2+9y^2=1$ at $P(\alpha ,\beta )$ is
$\Rightarrow T=0\Rightarrow 4\alpha x+9\beta y-1=0$
Given, above line is parallel to $y=\frac{8x}{9}+200.$
$\Rightarrow -\frac{4\alpha }{9\beta }=\frac{8}{9}$
$\Rightarrow \alpha =-2\beta$

Also, $P$ lies on the ellipse
$\Rightarrow 4{\alpha }^2+9{\beta }^2=1$
$\Rightarrow 4\times 4{\beta }^2+9{\beta }^2=1$
$\Rightarrow \beta =\pm \frac{1}{5}\Rightarrow \alpha =\mp \frac{2}{5}$
$\therefore |\alpha +\beta |=\frac{1}{5}$