Show that the angle between the tangent at any point P and the line joining P to the origin O is the same at all points on the curve log (x 2 + y 2 ) = k tan –1 
Text Solution
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Sol. We have,
log (x 2 + y 2 ) = k tan –1 
Differentiating w.r.t. x, we get
= k 
⇒ 2 
= k 
⇒ 2x + ky = (kx –2y) 
⇒ 
= 
Let the coordinates of P be (x 1 , y 1 ). Then
= 
If the tangent at P makes an angle θ with x- axis, then
tan θ = 
Suppose OP makes an angle φ with x- axis. Then,
tan φ = Slope of OP = 
Let α be the angle between OP and PT. Then,
θ = α + φ
⇒ α = θ – φ
⇒ tan α = tan ( θ – φ )
⇒ tan α = 
⇒ tan α = 
⇒ tan α = 
= 
⇒ α = tan –1 
= constant
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