A straight line through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed, then the locus of $R$ is:
- 3x + 2y = 6
- 2x + 3y = xy
- 3x + 2y = xy
- 3x + 2y = 6xy
The Correct Option is C
Solution and Explanation
(i) passes through the fixed point $(2,3)$
$\Rightarrow \frac{2}{a}+\frac{3}{b}=1$
$P(a, 0), Q(0, b), O(0,0)$, Let $R(h, k)$
Midpoint of $O R$ is $\left(\frac{h}{2}, \frac{k}{2}\right)$
Midpoint of $P Q$ is $\left(\frac{a}{2}, \frac{b}{2}\right) $
$\Rightarrow h=a, \,\,\,\,\, k=b \,\,\,\,\, \ldots$ (iii)
From (ii) & (iii),
$\frac{2}{h}+\frac{3}{k}=1 \,\,\,\,\, \Rightarrow$ locus of $R(h, k)$
$\frac{2}{x}+\frac{3}{y}=1 \,\,\,\,\, \Rightarrow 3 x+2 y=x y$
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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