Find the equation of line through (2,3) making 45° with x-axis

Question

Find the equation of a straight line passing through the point (2, 3) and making an angle of 45° with the positive x-axis.

Solution — Step by Step

The inclination of a line is the angle $\theta$ it makes with the positive x-axis, measured anticlockwise.

The slope (gradient) $m$ is related to the angle by:

m = \tan\theta

Given $\theta = 45°$ :

m = \tan 45° = 1

We know:

The line passes through point $(x_1, y_1) = (2, 3)$
Slope $m = 1$

The point-slope form of a line:

y - y_1 = m(x - x_1)

Substituting $(x_1, y_1) = (2, 3)$ and $m = 1$ :

y - 3 = 1 \times (x - 2)

y - 3 = x - 2

y = x - 2 + 3

\boxed{y = x + 1}

Or equivalently: $x - y + 1 = 0$

Check that (2, 3) lies on $y = x + 1$ : $3 = 2 + 1 = 3$ ✓

Check the angle: slope = 1 = tan 45° ✓

Final answer: $y = x + 1$ or $x - y + 1 = 0$

Why This Works

The slope of a line is defined as $m = \tan\theta$ where $\theta$ is the angle of inclination. For $\theta = 45°$ , $\tan 45° = 1$ exactly.

A slope of 1 means: for every 1 unit you move right along the x-axis, the line rises 1 unit. This gives a perfectly diagonal line (45° — as steep as it is flat). The specific line through (2,3) with this slope is uniquely determined by the point-slope formula.

The formula $y - y_1 = m(x - x_1)$ comes directly from the definition of slope: $m = (y - y_1)/(x - x_1)$ , rearranged.

Alternative Method — Slope-Intercept Form

Since $m = 1$ , the line has equation $y = x + c$ for some constant $c$ .

Substitute $(2, 3)$ : $3 = 2 + c \Rightarrow c = 1$ .

Therefore: $y = x + 1$ . Same answer, slightly different route.

Common Mistake

Confusing inclination (angle with x-axis) with angle between two lines. Here, the 45° is the angle the line makes with the positive x-axis, measured anticlockwise. So $m = \tan 45° = 1$ directly. Some students calculate the angle incorrectly when the angle is given as “with the y-axis” (that would be 90° − 45° = 45° in this symmetric case, but not in general). Always check whether the angle is measured from the x-axis or the y-axis.

Common inclinations to memorise: $\theta = 0° \Rightarrow m = 0$ (horizontal), $\theta = 30° \Rightarrow m = 1/\sqrt{3}$ , $\theta = 45° \Rightarrow m = 1$ , $\theta = 60° \Rightarrow m = \sqrt{3}$ , $\theta = 90° \Rightarrow m = \infty$ (vertical). These appear in almost every straight-lines chapter in Class 11.

Question

Solution — Step by Step

Why This Works

Alternative Method — Slope-Intercept Form

Common Mistake

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