Question

The radius vector of a point A relative to the origin varies as $\overrightarrow r = at\widehat i + b{t^2}\widehat j$ where a and b are positive constants. Find the equation of trajectory.
A.$y = \dfrac{b}{{{a^2}}}{x^2}$
B.$y = \dfrac{{{a^2}}}{b}x$
C.$y = \dfrac{{{a^2}}}{b}{x^2}$
D.None of these

Answer 1

In this question we have to find the equation of trajectory of point A. Radius vector of point A is given in the question. For this we will compare the given radius vector with the general radius vector and we will find the values of x and y then we will find the value of t and we will put the value of t in y to find the equation of trajectory.

Complete step by step answer:
Given,
Radius vector of point A relative to the origin is given by following equation
$\overrightarrow r = at\widehat i + b{t^2}\widehat j$…… (1)
Now comparing equation (1) with following general equation of position;
$\overrightarrow r = x\widehat i + y\widehat j$
It will give following values of x and y,
$x = at$…… (2)
And
$y = b{t^2}$……. (3)
Now we will find the value of t from equations (2)
$t = \dfrac{x}{a}$
Now we will put the value of t in equation (3) to get the equation of trajectory of point A,
$y = b{t^2}$
$y = b{\left( {\dfrac{x}{a}} \right)^2}$
$y = \dfrac{b}{{{a^2}}}{x^2}$
Result- Hence, from the above calculation we have found the trajectory equation of point A equal to$y = \dfrac{b}{{{a^2}}}{x^2}$.
Hence, option (A) is correct.

Note:
As we have seen in the above solution these types of questions are easy but the method to solve the question should be clear only then we will be able to do such questions. We might get a bit of a different case of this type of question. In that condition we will analyze the situation wisely then we can solve the question. One more thing that we must know is what is trajectory; trajectory is the curved path that an object follows when it is in motion with respect to time.