Grade 11Analytical Geometry

a variable tangent to the ellipse 4x^2+6y^2=12 intersects the co ordinate axes at A,B the curve described by the mid point of ab is

Shraddha Singh

10 Years agoGrade 11

1 Answer

BALAJI ANDALAMALA

10 Years ago

The equation of the given ellipse is $\frac{x^2}{3}+\frac{y^2}{2}=1$

Let $(\sqrt{3}cos\theta,\sqrt{2}sin\theta)$ be any point on the ellipse.

The equation of the tangent at this point is $\sqrt{3}cos\theta x+\sqrt{2}sin\theta y=1$
$A(\frac{1}{\sqrt{3}cos\theta},0) \,\,and\,\,B(0,\frac{1}{\sqrt{2}sin\theta})$
Let the mid point of AB is P(h,k).

$h=\frac{1}{2\sqrt{3}cos\theta} \,\,and\,\,k=\frac{1}{2\sqrt{2}sin\theta}$

$cos\theta=\frac{1}{2\sqrt{3}h} \,\,and\,\,sin\theta=\frac{1}{2\sqrt{2}k}$
$cos^2\theta+sin^2\theta=\left (\frac{1}{2\sqrt{3}h} \right )^2 +\left (\frac{1}{2\sqrt{2}k} \right )^2 =1$
The equation of the required curve is $\frac{1}{12x^2} +\frac{1}{8y^2} =1$