Write the equation of the lines for which $ an heta = \dfrac{1}{2}$, where $ heta$ is the inclination of the line and

(i) y-intercept is $-\dfrac{3}{2}$

(ii) x-intercept is $4$.

Question

Write the equation of the lines for which $	an	heta = \dfrac{1}{2}$, where $	heta$ is the inclination of the line and

(i) y-intercept is $-\dfrac{3}{2}$

(ii) x-intercept is $4$.

Accepted Answer

> [!TLDR]
> The slope of a line is $	an	heta$, so here $m = \frac12$. Using the slope-intercept form $y = mx + c$ and the intercept form $\frac{x}{a} + \frac{y}{b} = 1$, we get the two equations: $y = \frac12 x - \frac32$ and $x - 2y = 4$.

## Concept and Intuition

The inclination $	heta$ of a line is the angle it makes with the positive $x$-axis, measured anticlockwise. The slope $m$ is defined as $	an	heta$. So when you're told $	an	heta = \frac12$, you're directly given the slope: $m = \frac12$.

Now, a line is uniquely determined by its slope and one point on it. Here, instead of a point, we're given an intercept — either the $y$-intercept or the $x$-intercept. That's enough: the $y$-intercept is the $c$ in $y = mx + c$, and the $x$-intercept is the $a$ in $\frac{x}{a} + \frac{y}{b} = 1$ (where $b$ is the $y$-intercept). We just plug in. …

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Q.Write the equation of the lines for which $\tan\theta = \dfrac{1}{2}$ , where $\theta$ is the inclination of the line and

Unlock the full solution

Q.Write the equation of the lines for which $\tan\theta = \dfrac{1}{2}$ , where $\theta$ is the inclination of the line and

Unlock the full solution