Question:

The line joining $(5,0)$ to $((10 \cos \theta, 10 \sin \theta)$ is divided internally in the ratio $2: 3$ at $P$. If $q$ varies, then the locus of $P$ is

Updated On: Jun 27, 2024
  • a pair of straight lines
  • a circle
  • a straight line
  • None of these
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The Correct Option is B

Solution and Explanation

Let $P(x, y)$ be the point dividing the join of $A$ and $B$ in the ratio $2: 3$ internally, then $x=\frac{20 \cos \theta+15}{5}=4 \cos \theta+3 $ $\Rightarrow \cos \theta=\frac{x-3}{4} \ldots$..(i) $y=\frac{20 \sin \theta+0}{5}=4 \sin \theta $ $\Rightarrow \sin \theta=\frac{y}{4} \ldots$ (ii) Squaring and adding (i) and (ii), we get the required locus $(x-3)^{2}+y^{2}=16$, which is a circle.
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Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points. 

A straight line can be represented as an equation in various forms,  as show in the image below:

 

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y0 = m (x – x0)

2. Two – Point Form

Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2)  are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes

The slope of P2P = The slope of P1P2 , i.e.

\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)

Hence, the equation becomes:

y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c