If the image of point $P(2, 3)$ in a line $L$ is $Q(4,5)$, then the image of point $R(0,0)$ in the same line is:
- $\left(2,2\right)$
- $\left(4,5\right)$
- $\left(3,4\right)$
- $\left(7,7\right)$
The Correct Option is D
Solution and Explanation
Slope of $PQ = 1$
Slope of the line $L = -1$
Mid-point $(3,4)$ lies on the line $L$. Equation of line $L$,
$y-4=-1 \left(x-3\right) \Rightarrow x+y -7=0 $ ...(i)
Let image of point $R(0,0)$ be $S$ $\left(x_{1}, y_{1}\right)$
Mid-point of $RS = \left(\frac{x_{1}}{2}\frac{y_{1}}{2}\right)$
Mid-point $\left(\frac{x_{1}}{2},\frac{y_{1}}{2}\right)$ lies on the line (i)
$\therefore x_{1}+y_{1}=14$
Slope of RS =$\frac{y_{1}}{x_{1}}$
Since RS $\bot$ line L
$\therefore \frac{y_{1}}{x_{1}} \times\left(-1\right)=-1$
$\therefore x_{1} =y_{1}$
From (ii) and (iii),
$x_{1} =y_{1}=7$
Hence the image of $R = (7,7)$
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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
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