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Question

Number of normals at the points on the curve y=x(1x2) where the tangents makes an angle of π4 with x-axis

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Solution

Given eqn of curve is
y=x1x2
Also, given slope of tangent to curve is tanπ4=1
dydx=1.(1x2)(2x)x(1x2)2
1=1+x2(1x2)2
(1+x2)=(1x2)2
x43x2=0
x=0,3,3
At x=0y=0
At x=3y=32
At x=3y=32
Hence the points are : (0,0),(3,32)and(3,32).
Eqn of normal at (0,0) is
y0=1(x0)
x+y=0
Eqn of normal at (3,32) is
y+32=1(x3)
2y+2x=3
Eqn of normal at (3,32) is
y32=1(x+3)
2y+2x=3

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