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 $\ln (x^2+y^2)=c{\tan }^{-1}\frac{y}{x},c$ is constant, is

• A. Independent of $x$
• B. Independent of $y$
• C. Independent of $x$ but dependent on $y$
• D. Independent of $y$ but dependent on $x$

Answer 1

Answer: (A), (B)

Independent of $y$

Let $P(x,y)$ be a point on the curve $\ln (x^2+y^2)=c{\tan }^{-1}\frac{y}{x}.$
Differentiating both sides with respect to $x$, we get
$\frac{2x+2y{y}^{\prime }}{(x^2+y^2)}=\frac{c(x{y}^{\prime }-y)}{(x^2+y^2)}$
$\Rightarrow y^{\prime }=\frac{2x+cy}{cx-2y}=m_1$ (say)

Slope of $OP=\frac{y}{x}=m_2$ (say) (where $O$ is origin)

Let the angle between the tangents at $P$ and $OP$ be $\theta .$ Then,
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$
$=\left|\frac{\frac{2x+cy}{cx-2y}-\frac{y}{x}}{1+\frac{2xy+c{y}^2}{c{x}^2-2xy}}\right|$
$=\left|\frac{2}{c}\right|$
$\Rightarrow \theta ={\tan }^{-1}\left(\frac{2}{c}\right)$
which is independent of $x$ and $y.$