Find the equation of a line drawn perpendicular to the line \(\frac{x}{4} +\frac{ y}{6} = 1 \) through the point where it meets the y-axis
Solution and Explanation
The equation of the given line is \(\frac{ x}{4} +\frac{ y}{6} = 1\)
This equation can also be written as \(3x + 2y - 12 = 0\)
\(y =\frac{ -3}{2} x + 6\), which is of the form \(y = mx + c\)
∴ Slope of the given line \(=-\frac{3}{2}\)
∴ Slope of line perpendicular to the given line \(=\frac{-1}{(\frac{-3}{2})} = \frac{2}{3}\)
Let the given line intersect the y-axis at (0, y).
On substituting x with 0 in the equation of the given line, we obtain
\(\frac{y}{6} = 1\)
\(⇒y = 6\)
∴ The given line intersects the y-axis at (0, 6).
The equation of the line that has a slope of \(\frac{2}{3}\) and passes through point (0, 6) is
\((y – 6) = \frac{2}{3} (x – 0)\)
\(3y – 18 = 2x\)
\(2x – 3y + 18 = 0\)
Thus, the required equation of the line is \(2x – 3y + 18 = 0\)
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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