Question

Given a curve $C$. Suppose that the tangent line at $P(x,y)$ on $C$ is perpendicular to the line joining $P$ and $Q(1,0)$. If the line $2x+3y-15=0$ is tangent to the curve $C$ then the curve $C$ denotes.

• A. a circle touching the $x$-axis.
• B. a circle touching the $y$-axis.
• C. circle whose $y$-intercept is $4\sqrt{3}$
• D. a parabola with axis parallel to y-axis.

Answer 1

Answer: (C)

$Y-y=m(X-x)$
$\text{ now, }\left(\frac{y-0}{x-1}\right)m=-1$
$y\frac{dy}{dx}=1-x$
$\text{ integrating }\frac{y^2}{2}=x-\frac{x^2}{2}+C$
$x^2+y^2-2x=C$ ....(1)
This is the equation of a circle with centre $(1,0)$
$\therefore 2x+3y-15=0$ is tangent at (1)
$\therefore$ perpendicular from $(1,0)$ on the line $=r$
$\left|\frac{2-15}{\sqrt{13}}\right|=\sqrt{1+C}$
$1=13\Rightarrow C=12$
hence the curve is $x^2+y^2-2x=12$