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

Concept: Slope Calculation — The slope $m$ of a line is $\tan\theta$, where $\theta$ is the angle of inclination. Here $m = \frac{1}{2}$.

Step 1: For part (i), use the slope-intercept form $y = mx + c$.

Given $m = \frac{1}{2}$ and $c = -\frac{3}{2}$, the equation is

Step 2: Multiply through by 2 to clear fractions:

Step 3: For part (ii), use the intercept form $\frac{x}{a} + \frac{y}{b} = 1$ with $a = 4$. …

Concept and Intuition

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

Step-by-step solution

(i) $y$-intercept is $-\frac32$

1. Identify the slope.

   Since $\tan\theta = \frac12$, we have $m = \frac12$.

2. Use the slope-intercept form.

   The general equation is $y = mx + c$, where $c$ is the $y$-intercept.

   Here $c = -\frac32$, so:

3. Optional: rewrite in standard form. Multiply through by 2:

Rearranging:

So the line is $y = \frac12 x - \frac32$ (or equivalently $x - 2y = 3$).

(ii) $x$-intercept is $4$

1. Slope is still $m = \frac12$.

2. Use the intercept form.

   The intercept form of a line is $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ is the $x$-intercept and $b$ is the $y$-intercept.

   Here $a = 4$, so:

3. Find $b$ using the slope. The slope in intercept form can be found by rewriting in $y = mx + c$ form. Solve for $y$: …