Question

The equations of the tangents to the ellipse $9x^2+16y^2=144$ from the point (2,3) are

• A. y=3, x=5
  
• B. x=2, y=3
  
• C. x=3, y=2
  
• D. x+y=5, y=3

Answer 1

The correct option is D x+y=5, y=3

$9x^2+16y^2=144$
$\Rightarrow \frac{x^2}{16}+\frac{y^2}{9}=1$
Equation of the tangent in case of an ellipse is given by
$y=mx+\sqrt{a^2{m}^2+b^2}$
$\Rightarrow y=mx+\sqrt{16m^2+9}$ ...(1)
Substituting x=2 and y=3, we get:
$3=2m\pm \sqrt{16m^2+9}$
$3-2m=\sqrt{16m^2+9}$
On squaring both sides, we get
$(3-2m{)}^2=(16m^2+9)$
$\Rightarrow 9+4m^2-12m=(16m^2+9)$
$\Rightarrow 12m^2+12m=0$
$\Rightarrow 12m(m+1)=0$
$\Rightarrow m=0,-1$
Substituting values of m in eq. (1), we get:
For m=0, y=3
For m=-1, y=-x+5 or x+y=5