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Question

If the tangent at (x1,y1) to the curve x3+y3=a3 meets the curve again in (x2,y2), and x2x1+y2y1=P then the value of |P| is 


Solution

Slope of tangent at (x1,y1) is x21y21
The tangent cuts the curve again at (x2,y2). Therefore, 
slope of the tangent =y2y1x2x1
x21y21=y2y1x2x1       (1)
Also, x31+y31=a3 and x32+y32=a3
x31+y31=x32+y32
y32y31x31x32=1
y2y1x1x2=x21+x22+x1x2y21+y22+y1y2
x21y21=x21+x22+x1x2y21+y22+y1y2
x21y21+x21y22+x21y1y2=y21x21+x22y21+y21x1x2
x22y21y22x21=x1y1[x1y2x2y1]
x2y1+y2x1=x1y1
x2x1+y2y1=1

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