If the tangent at any point of the curve x^2/3+y^2/3=a^2/3 meets the axes of coordinates in A and B then prove that the locus of the midpoint of AB is a circle.

Question

If the tangent at any point of the curve x^2/3+y^2/3=a^2/3 meets
the axes of coordinates in A and B then prove that the locus of the
midpoint of AB is a circle

Rahul Raj · Accepted Answer

dear student

any point on the curve can be taken as(acos3θ,asin3θ)slope of tangent on differentiating curve23x1/3+23y1/3dydx=0dydx=-y1/3x1/3for the give point slope=-tanθequation of tangent is(y-asin3θ)=(x-acos3θ)(-tanθ)simplifying we getxtanθ+y=asinθxsinθ+ycosθ=asinθcosθit will cut axes at (acosθ,0) and (0,asinθ)midpoint=(acosθ/2,asinθ/2)let it be (h,k)thencosθ=2h/asinθ=2k/acos2θ+sin2θ=14h2+4k2=a2locus isx2+y2=a2/4 which is circle

regards