Question

The trajectories orthogonal to $x^2+y^2=2ax$ is

• A. $y=c(x-a)$
• B. $y=c(x^2+y^2)$
• C. $y=c(x^2+2y^2)$
• D. $y^2=x$

Answer 1

The correct option is A $y=c(x-a)$

$x^2+y^2=2ax$

differentiate equation with respect to $x$

$2x+2y\frac{dy}{dx}=2a$

$\frac{dy}{dx}=\frac{a-x}{y}$

this is the slope of tangent to curve slope of tangent perpendicular $=\frac{-1}{\left(dy/dx\right)}$

thus for new curve $\frac{dy}{dx}=\frac{-y}{(a-x)}$

$\frac{dy}{y}-\frac{dx}{x-a}\Rightarrow \ln y=\ln (x-a)+\ln c$

$y=C(x-a)$