Question:

If the angle between two lines is $\frac{\pi}{4}$ and slope of one of the lines is $\frac{1}{2}$, find the slope of the other line.

Updated On: Jul 6, 2022

$3$ or $\frac{-1}{3}$
$2$ or $\frac{-1}{2}$
$4$ or $\frac{-1}{4}$
$3$ or $-3$

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The Correct Option is A

Solution and Explanation

We know that the acute angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is given by $tan\,\theta=\left|\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}\right|\quad\ldots\left(i\right)$ Let $m_{1}=\frac{1}{2}$, $m_{2}=m$ and $\theta=\frac{\pi}{4}$. Now, putting these values in $\left(i\right)$, we get $tan \frac{\pi}{4}=\left|\frac{m-\frac{1}{2}}{1+\frac{1}{2}m}\right|$ $\Rightarrow 1=\left|\frac{2m-1}{2+m}\right|$, wihich gives $\frac{2m-1}{2+m}=1$ or $\frac{2m-1}{2+m}=-1$ Therefore $m = 3$ or $m=-\frac{1}{3}$ Hence, slope of the other line is $3$ or $-\frac{1}{3}$.

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Notes on Straight lines

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