The perpendicular from the origin to the line \(y = mx + c\) meets it at the point \((-1, 2)\). Find the values of m and c.
Solution and Explanation
The given equation of line is \(y = mx + c.\)
It is given that the perpendicular from the origin meets the given line at \((-1, 2). \)
Therefore, the line joining the points\( (0, 0)\) and \((-1, 2)\) is perpendicular to the given line.
∴ Slope of the line joining\( (0, 0)\) and \((-1, 2)=\frac{2}{-1}=-2\)
The slope of the given line is m.
\(∴ m\times-2=-1\) [The two lines are perpendicular]
\(⇒m=\frac{1}{2}\)
Since point \( (-1, 2)\) lies on the given line, it satisfies the equation \(y = mx + c.\)
\(∴ 2=m(-1)+c\)
\(⇒ 2=\frac{1}{2}(-1)+c\)
\(⇒ c = 2+\frac{1}{2} =\frac{ 5}{2}\)
Thus, the respective values of m and c are \(\frac{1}{2}\) and \(\frac{5}{2}\), respectively.
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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