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Question

Let x2/3+y2/3=a2/3,(a>0) be the equation of curve then which of the following is/are correct?

A
Equation of tangent at (x1,y1) on the curve is xx1/31+yy1/31=a2/3
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B
Length of portion of tangent intercepted between the coordinate axes is constant.
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C
If the curve x2/3+y2/3=a2/3 touches the curve x2c2+y2d2=1 then c+d=a
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D
Equation of normal at (x1,y1) on the curve is x.x1/31y.y1/31=x4/31+y4/31
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Solution

The correct options are
A Equation of tangent at (x1,y1) on the curve is xx1/31+yy1/31=a2/3
B Length of portion of tangent intercepted between the coordinate axes is constant.
C If the curve x2/3+y2/3=a2/3 touches the curve x2c2+y2d2=1 then c+d=a
(x,y)=(acos3θ,asin3θ)
dydx=sinθcosθ
Equation of tangent (yasin3θ)=sinθcosθ(xacos3θ)
xcosθ+ysinθ=a
xx1/31+yy1/31=a2/3


Length of the portion of the tangent between the coodinate axes : AB=a2cos2θ+a2sin2θ=a

Tangent at (x1y1) to x2c2+y2d2=1 is
xx1c2+yy1d2=1
which is identical to xx1/31+xy1/31=a2/3
So x1c2.×x1/31=y1cl2×y1/31=1a2/3
x4/31=c2a2/3, y4/31=d2a2/3
x2/31=ca1/3, y2/31=da2/3

which lies on x2/3+y2/3=a2/3
ca1/3+da1/3=a2/3
c+d=a

Equation of normal
y=y1/3x1/3
So normal is
(yy1)=x1/31y1/3(xx1)

xx1/31yy1/31=x4/31y4/31

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