Let the equation of a curve be 4 x 2 + 3 y 2 = 1 where ( 2 cos θ , 3 sin θ ) is a general point on the curve. If the tangent to the given curve intersects the co-ordinate axes at points A , B ,

Question

Let the equation of a curve be 4 x 2 ​ + 3 y 2 ​ = 1 where ( 2 cos θ , 3 ​ sin θ ) is a general point on the curve. If the tangent to the given curve intersects the co-ordinate axes at points A , B , then the locus of midpoint of AB is

TestHub · Accepted Answer

Equation of tangent is y − 3 ​ sin θ = − 2 3 ​ ​ cot θ ( x − 2 cos θ ) ⇒ x-intercept ( x 0 ​ ) = c o s θ 2 ​ , y-intercept ( y 0 ​ ) = s i n θ 3 ​ ​ . If mid point is ( h , k ) : h = 2 x 0 ​ ​ , k = 2 y 0 ​ ​ , so h 2 1 ​ + 4 k 2 3 ​ = 1 ⇒ 3 h 2 + 4 k 2 = 4 h 2 k 2 . Locus is 3 x 2 + 4 y 2 = 4 x 2 y 2

Mathematics - Application of Derivative Question with Solution | TestHub

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