Question

The angle between the tangents at any point P and the line joining P to origin O, where P is a point on the curve ln

(x² + y²) = c tan^–1 y/x, c is a constant, is

• A. Constant
• B. Varies at tan^–1 (x)
• C. Varies as tan^–1(y)
• D. None of these

Answer 1

Answer: (A)

Constant

Answer 2

P(x, y) be a point on the curve

ln (x² + y²) = c tan^–1 y/x

differentiating both side with respect to x

$\frac{2x + 2yy'}{(x^{2} + y^{2})} = \frac{c(xy' - y)}{x^{2} + y^{2}}$ ⇒ y ' = $\frac{2x + cy}{cx - 2y}$ = m₁

slope of OP = y/x = m₂

So tan θ = $\left| \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}} \right|$ = $\left| \frac{\frac{2x + cy}{cx - 2y} - \frac{y}{x}}{1 + \frac{2xy + cy^{2}}{cx^{2} - 2xy}} \right|$ = 2/c

θ = tan^–1 (2/c) which is independent of x and y