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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

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

Grade:11

1 Answers

BALAJI ANDALAMALA
askIITians Faculty 78 Points
5 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

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