Question

The position of a moving point in the XY-plane at time t is given by $\left( (u\cos\alpha)t,(u\sin\alpha)t - \frac{1}{2}gt^{2} \right),$ where $u,\alpha,g$are constants. The locus of the moving point is.

• A. A circle
• B. A parabola
• C. An ellipse
• D. None of these

Answer 1

Answer: (B)

A parabola

Answer 2

Let then $t = \frac { h } { u \cos \alpha }$. Putting the value of t in $k = u \sin \alpha \cdot t - \frac { 1 } { 2 } g t ^ { 2 }$ we get $k = h \tan \alpha - \frac { 1 } { 2 } g \frac { h ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$

$\therefore$Locus of (h, k) is $y = x \tan \alpha - \frac { 1 } { 2 } g \frac { x ^ { 2 } } { u ^ { 2 } \cos ^ { 2 } \alpha }$, which is a parabola.