Question

The curve which passes through $(1,2)$ and whose tangent at any point has a slope that is half of slope of the line joining origin to the point of contact, is -

• A. A rectangle hyperbola
• B. A circle
• C. A parabola
• D. A straight line through origin
• E. Answer required

Answer 1

The correct option is C A parabola

Let the arbitrary point be $x,y.$
Now the slope of the line joining the origin and the point is
$=\frac{y-0}{x-0}$
$=\frac{y}{x}$.
The equation of the curve
$y=f\left(x\right)$.
Now slope of the tangent
$=\frac{dy}{dx}$
$=\frac{y}{2x}$ ... as per the given condition.
Hence
$2\frac{dy}{y}=\frac{dx}{x}$
Integrating both sides we get
$2\ln y=\ln x+\ln c$
Now $y_{x=1}=2$
Hence
$2\ln \left(2\right)=\ln \left(1\right)+\ln \left(c\right)$
Or
$c=2^2=4$
Hence
$2\ln y=\ln x+\ln 4$
Or
$\ln \left(y^2\right)=\ln 4x$
Or
$y^2=4x$ is the required equation.
This is an equation of parabola.