Question

The locus of the midpoint of the portion of the tangent to the ellipse x²/a² +  y²/b² = 1 included between the axes is the curve a²/x² + b²/y² = k where k is equal to ........

Answer 1

The tangent at (a cos θ, b sin θ)  to the ellipse is

x/a cosθ + y/b sin θ = 1

This meets the axes at the points A(a sec θ , 0) and B(0, b cosec θ). Let (x₁, y₁) be the midpoint of  (bar)AB. Therefore

Therefore, the locus of (x₁, x₂) is

a²/x² + b²/y² = 4