Find the image of the point (3, 8) with respect to the line x + 3y = 7 assuming the line to be a plane mirror.
Solution and Explanation
The equation of the given line is
\(x + 3y = 7 … (1) \)
Let point B (a, b) be the image of point A (3, 8).
Accordingly, line (1) is the perpendicular bisector of AB.
Slope of\( AB =\frac{b-8}{a-3}\), while the slope of the line \((l)=-\frac{1}{3}\)
Since line (1) is perpendicular to AB,
\(\left(\frac{b-8}{a-3}\right)\times\left(\frac{-1}{3}\right)=-1\)
\(⇒\frac{ b-8}{3a-9}=1\)
\(⇒ b-8=3a-9\)
\(⇒ 3a-b=1 ....(2)\)
Mid-Point of \(AB =\left(\frac{a+3}{2},\frac{b+8}{2}\right)\)
The mid-point of line segment AB will also satisfy line (1).
Hence, from equation (1), we have
\(\left(\frac{a+3}{2}\right)+3\left(\frac{b+8}{2}\right)=7\)
\(⇒ a+3+3b+24=14\)
\(⇒ a+3b=-13 ......(3)\)
On solving equations (2) and (3), we obtain a = -1 and b = -4.
Thus, the image of the given point with respect to the given line is (-1, -4).
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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