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. Then find the type of the curve.
A. Circle
B. Ellipse
C. Parabola
D. Hyperbola

Answer 1

Hint First, rewrite the given condition of the slope in the equation form. Then integrate the equation with respect to that variable. After that use logarithmic properties to simplify the equation. In the end, equate the equation with standard equations of the curve to get the required answer.

Formula used
\[\int {\dfrac{1}{x}} dx = \log x\]
\[m\log a = \log {a^m}\]
\[\log a + \log b = \log\left( {ab} \right)\]
If \[\log a = \log b\], then \[a = b\]

Complete step by step solution:
It is given 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.
Let \[A\left( {x,y} \right)\] be the point on the curve and \[\dfrac{{dy}}{{dx}}\] be the slope of the tangent.
Let’s simplify the given condition.
\[\dfrac{{dy}}{{dx}} = 2\left( {\dfrac{{y - 0}}{{x - 0}}} \right)\]
\[ \Rightarrow \]\[\dfrac{{dy}}{{dx}} = 2\left( {\dfrac{y}{x}} \right)\]
\[ \Rightarrow \]\[\dfrac{1}{y}dy = \dfrac{2}{x}dx\]
Now take the integral of both sides.
\[\int {\dfrac{1}{y}dy} = \int {\dfrac{2}{x}dx} \]
\[ \Rightarrow \]\[\log y = 2\log x + \log C\] , where \[C\] is an integration constant.
Apply the logarithmic property \[m\log a = \log {a^m}\].
\[\log y = \log {x^2} + \log C\]
Now apply the property \[\log a + \log b = \log\left( {ab} \right)\].
\[\log y = \log C{x^2}\]
\[ \Rightarrow \]\[y = C{x^2}\]
Clearly, it is an equation of a parabola that is open in the direction of the positive y-axis.
Hence the correct option is C.

Note: A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.
The standard equations of the parabola are:
\[{y^2} = 4ax\], at positive x-axis
\[{y^2} = - 4ax\], at negative x-axis
\[{x^2} = 4ay\], at positive y-axis
\[{x^2} = - 4ay\], at negative y-axis