Question

A curve having the condition that the slope of the tangent at some point is two times the slope of the straight line joining the same point to the origin of coordinates is a/an

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

Answer 1

The correct option is C

parabola

Explanation for the correct option:

Step 1: Find the slope of the straight line joining the point to the origin of coordinates.

Let $P(x,y)$be any point on the curve.

the slope of the straight line joining the point $P(x,y)$ to the origin of coordinates would be,

$m=\frac{y-0}{x-0}$

Step 2: Formulate the given problem into mathematical expressions.

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=2m \\ \Rightarrow \frac{dy}{dx}=2\frac{y}{x} \end{gathered}$$

Step 3: Find the equation of the curve.

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=2\frac{y}{x} \\ \Rightarrow \frac{1}{y}dy=2\frac{1}{x}dx \\ \Rightarrow \int \frac{1}{y}dy=\int 2\frac{1}{x}dx \\ \Rightarrow \log \left(y\right)=2\log \left(x\right)+\log \left(C\right) \\ \Rightarrow y=C{x}^2 \end{gathered}$$

This curve represents a parabola.

Therefore, option (C) is the correct answer.