(B) Let $x = (u \cos \alpha)t$ and $y = (u \sin \alpha)t - \frac{1}{2}gt^2$.From the first equation,we have $t = \frac{x}{u \cos \alpha}$.Substituting

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(B) Let $x = (u \cos \alpha)t$ and $y = (u \sin \alpha)t - \frac{1}{2}gt^2$.From the first equation,we have $t = \frac{x}{u \cos \alpha}$.Substituting this value of $t$ into the equation for $y$:$y = (u \sin \alpha) \left( \frac{x}{u \cos \alpha} \right) - \frac{1}{2}g \left( \frac{x}{u \cos \alpha} \right)^2$$y = x \tan \alpha - \frac{g x^2}{2 u^2 \cos^2 \alpha}$.This equation is of the form $y = ax^2 + bx + c$,which represents a parabola.

Class 11 Mathematics 2 Conic Sections: Parabola, Ellipse and Hyperbola Parabola

Medium

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

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Class 11 Questions

Class 11

Mathematics Questions

Chapter

2 Conic Sections: Parabola, Ellipse and Hyperbola

Subtopic

Parabola

The line $y - \sqrt{3}x + 3 = 0$ cuts the parabola $y^2 = x + 2$ at the points $P$ and $Q$. If the coordinates of the point $X$ are $(\sqrt{3}, 0)$, then the value of $XP \cdot XQ$ is

MediumWBJEE 2025

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Find the length of the latus rectum of the parabola $y = x \tan \alpha - \frac{g x^2}{2 u^2 \cos^2 \alpha}$.

Medium

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What is the distance between the focus and the directrix of the parabola $x^2 = -8y$?

Easy

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The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $x$-axis is

DifficultIIT 2001

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$A$ parabola has its focus on the positive $X$-axis and the $Y$-axis as its directrix. If $P(\alpha, 4)$ is a point on this parabola such that the tangent to the parabola at point $P$ passes through the origin, then the distance of $P$ from the origin is

DifficultMHT CET 2026

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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

Similar Questions

The line $y - \sqrt{3}x + 3 = 0$ cuts the parabola $y^2 = x + 2$ at the points $P$ and $Q$. If the coordinates of the point $X$ are $(\sqrt{3}, 0)$, then the value of $XP \cdot XQ$ is

Find the length of the latus rectum of the parabola $y = x \tan \alpha - \frac{g x^2}{2 u^2 \cos^2 \alpha}$.

What is the distance between the focus and the directrix of the parabola $x^2 = -8y$?

The equation of the common tangent touching the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $x$-axis is

$A$ parabola has its focus on the positive $X$-axis and the $Y$-axis as its directrix. If $P(\alpha, 4)$ is a point on this parabola such that the tangent to the parabola at point $P$ passes through the origin, then the distance of $P$ from the origin is

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