Question

The equation of tangents to the ellipse $9x^2+16y^2=144$ which pass through the point $(2,3)-$

• A. $3x+8y=30$
• B. $x+y=2$
• C. $3x-7y=30$
• D. $y=3;x+y=5$

Answer 1

The correct option is A $3x+8y=30$

$9x^2+16y^2=144$

$\Rightarrow \frac{x^2}{16}+\frac{y^2}{9}=1$

$\Rightarrow y^2=9(1-\frac{x^2}{16})$

$2y\left(\frac{dy}{dx}\right)=-\frac{9\left(2x\right)}{16}$

$\left[\frac{dy}{dx}\right]=-\frac{9x}{16y}\Rightarrow {\left(\frac{dy}{dx}\right)}_{(2,3)}\Rightarrow \frac{-9\left(2\right)}{16\times \left(3\right)}\Rightarrow \frac{-3}{8}$

equation of tangent

$\Rightarrow (y-y_1)=m(x-x_1)$

$m={\left(\frac{dy}{dx}\right)}_{(x,y)}\Rightarrow \frac{-3}{8}=m$

$\Rightarrow (y-3)=\frac{-3}{8}(x-2)$

$\Rightarrow 8y-24=-3x+6$

$\Rightarrow 3x+8y=30$