Equations of the lines through the point $(3,2)$ and making an angle of $45^\circ$ with the line $x-2y=3$ are ____.

Equations of the lines through the point (3,2) and making an angle of 45^° with the line x-2y=3 are __.

NCERT Exemplar · Q44

(3,2)

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Concept understanding — Slope Calculation

Slope Calculation — From Intuition to Precision

Imagine you're walking up a hill. Some hills are gentle — you barely notice the climb. Others are so steep you have to lean forward and use your hands. That "steepness" is what slope measures. In mathematics, slope tells us how fast a line rises or falls as we move from left to right.

The Intuition: Rise Over Run

Take any two points on a straight line. As you walk from the left point to the right point, two things happen:

You move horizontally — that's the run.
You move vertically — that's the rise (upwards) or fall (downwards).

Slope is simply the ratio:

Slope = (vertical change) ÷ (horizontal change)

If you climb 3 metres while walking 5 metres forward, the slope is $3/5 = 0.6$ . If you descend 2 metres while walking 4 metres forward, the slope is $-2/4 = -0.5$ — negative because you're going downhill.

The Precise Definition

Given two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a non-vertical line, the slope $m$ is:

m = \frac{y_2 - y_1}{x_2 - x_1}

The numerator is the rise (change in $y$ ), the denominator is the run (change in $x$ ). The order matters: subtract the first point's coordinates from the second's, consistently.

Watch out

Never divide by zero. If $x_2 = x_1$ , the line is vertical — slope is undefined (not zero, not infinite — just undefined).

What the Number Tells You

Slope value	What the line does
$m > 0$	Rises left to right (uphill)
$m < 0$	Falls left to right (downhill)
$m = 0$	Horizontal (flat)
$m$ undefined	Vertical (straight up/down)

The larger the absolute value $|m|$ , the steeper the line. A slope of $5$ is much steeper than a slope of $0.2$ .

A Worked Example

Find the slope of the line through $(1, 2)$ and $(4, 8)$ .

Step 1: Label the points. Let $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 8)$ .

Step 2: Compute the rise: $y_2 - y_1 = 8 - 2 = 6$ .

Step 3: Compute the run: $x_2 - x_1 = 4 - 1 = 3$ .

Step 4: Divide: $m = \frac{6}{3} = 2$ .

The line rises 2 units vertically for every 1 unit it moves right. …

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The key idea is to use the slope formula for the angle between two lines. The slopes of the required lines are found by solving $\tan 45^\circ = \left| \frac{m - \frac12}{1 + \frac12 m} \right|$ , giving $m = 3$ and $m = -\frac13$ . The equations through $(3,2)$ are $3x - y = 7$ and $x + 3y = 9$ .

Concept and Intuition

When a problem says "a line makes an angle of $45^\circ$ with another line," it is always about the acute angle between their directions — that is, the angle between their slopes. The formula connecting the angle $\theta$ between two lines with slopes $m_1$ and $m_2$ is:

\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|

Here, one line is given: $x - 2y = 3$ . Its slope is $m_1 = \frac12$ . We want lines through $(3,2)$ whose slope $m$ makes a $45^\circ$ angle with this line. So we set $\tan 45^\circ = 1$ and solve for $m$ .

Because the formula uses absolute value, we get two possible slopes — one steeper, one shallower — corresponding to the two lines that can be drawn through the point at exactly $45^\circ$ on either side of the given line.

Step-by-step solution

1. Find the slope of the given line.

Rewrite $x - 2y = 3$ as $y = \frac12 x - \frac32$ .

So the slope is $m_1 = \frac12$ .

2. Write the slope of the unknown line.

Let the required line through $(3,2)$ have slope $m$ . Its equation will be $y - 2 = m(x - 3)$ .

3. Apply the angle-between-lines formula.

We want $\theta = 45^\circ$ , so $\tan 45^\circ = 1$ .

Thus:

1 = \left| \frac{m - \frac12}{1 + \frac12 m} \right|

4. Remove the absolute value — two cases.

Case 1:

\frac{m - \frac12}{1 + \frac12 m} = 1

Multiply through: $m - \frac12 = 1 + \frac12 m$

$\Rightarrow m - \frac12 m = 1 + \frac12$

$\Rightarrow \frac12 m = \frac32$

$\Rightarrow m = 3$

Case 2:

\frac{m - \frac12}{1 + \frac12 m} = -1

Multiply: $m - \frac12 = -1 - \frac12 m$

$\Rightarrow m + \frac12 m = -1 + \frac12$

$\Rightarrow \frac32 m = -\frac12$

$\Rightarrow m = -\frac13$ …

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Q.Equations of the lines through the point (3,2)(3,2)(3,2) and making an angle of 45∘45^\circ45∘ with the line x−2y=3x-2y=3x−2y=3 are ____.

Concept and Intuition

Step-by-step solution

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Q.Equations of the lines through the point $(3,2)$ and making an angle of $45^\circ$ with the line $x-2y=3$ are ____.

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