Question

Find the curve possessing the property that the intercept, the tangent at any point of a curve cuts off on the y-axis is equal to the square of the abscissa of the point of tangency, the curve is a:

• A. line
• B. circle
• C. parabola
• D. ellipse

Answer 1

Answer: (C)

parabola

The equation of the tangent at any point $(x,y)$ is given by
$Y-y=\frac{dy}{dx}(X-x)$ when $X=0;Y=y-x\frac{dy}{dx}=y-$ intercept
It is given that $Y=x^2$
$\Rightarrow y-x\frac{dy}{dx}=x^2\Rightarrow \frac{dy}{dx}-\frac{y}{x}=-x$
$I.F.=e^{\int -\frac{1}{x}dx}=e^{-\log x}=\frac{1}{x}$
$\therefore$ the solution is
$y.\frac{1}{x}=\int -x.\frac{1}{x}dx=-x+c\Rightarrow y=cx-x^2$