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Question

The angle between the tangent at any point P and the line joining P to the origin, where P is a point on the curve ln(x2+y2)=ctan1yx,c is constant, is

A
Independent of x
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B
Independent of y
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C
Independent of x but dependent on y
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D
Independent of y but dependent on x
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Solution

The correct option is B Independent of y
Let P(x,y) be a point on the curve ln(x2+y2)=ctan1yx.
Differentiating both sides with respect to x, we get
2x+2yy(x2+y2)=c(xyy)(x2+y2)
y=2x+cycx2y=m1 (say)

Slope of OP=yx=m2 (say) (where O is origin)

Let the angle between the tangents at P and OP be θ. Then,
tanθ=m1m21+m1m2
=∣ ∣ ∣ ∣2x+cycx2yyx1+2xy+cy2cx22xy∣ ∣ ∣ ∣
=2c
θ=tan1(2c)
which is independent of x and y.

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