Question

What is the curve which passes through the point (1, 1) and whose slope is $\frac{2y}{x}$?

• A. Circle
• B. Parabola
• C. Ellipse
• D. Hyperbola

Answer 1

Answer: (B)

Parabola

Let the curve be $y=f\left(x\right)$,

$\therefore$ slope of the tangent drawn at any point on the curve is $\frac{df\left(x\right)}{dx}=f^{{}^{\prime }}\left(x\right)$,

Given that slope at any point on the curve is $\frac{2y}{x}$,

$⟹\frac{dy}{dx}=\frac{2y}{x}⟹\int \frac{1}{y}dy=\int \frac{2}{x}dx$

$⟹\ln y=2\ln x+c$

Where $c$ is the integration constant,

Given that the curve passes through the point $(1,1)$,

$⟹c=0$,

$\therefore y=x^2$ is the equation of the curve which is a parabola.