The equation of the line through the intersection of the lines $2x-3y=0$ and $4x-5y=2$ and — match Column $C_1$ with Column $C_2$. Column $C_1$: (a) through the point $(2,1)$ is; (b) perpendicular to the line $x+2y+1=0$ is; (c) parallel to the line $3x-4y+5=0$ is; (d) equally inclined to the axes is. Column $C_2$: (i) $2x-y=4$; (ii) $x+y-5=0$; (iii) $x-y-1=0$; (iv) $3x-4y-1=0$.

The equation of the line through the intersection – Class 11 Mathematics

NCERT Exemplar · Q59

2x-3y=0

(a) through the point

(2,1)

is;

(b) perpendicular to the line

x+2y+1=0

is;

3x-4y+5=0

is;

(d) equally inclined to the axes is. Column

C_2

(i)

2x-y=4

;

(ii)

x+y-5=0

;

(iii)

x-y-1=0

;

(iv)

3x-4y-1=0

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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 intersection point of the two given lines is found first. Then, using the family of lines through that point, each condition in Column C₁ gives a specific line equation, which is matched to Column C₂.

Step 1: Find the intersection point of the given lines

We solve:

2x - 3y = 0 \quad \text{and} \quad 4x - 5y = 2

From the first equation: $2x = 3y \Rightarrow x = \frac{3y}{2}$ .

Substitute into the second:

4\left(\frac{3y}{2}\right) - 5y = 2 \Rightarrow 6y - 5y = 2 \Rightarrow y = 2

Then $x = \frac{3(2)}{2} = 3$ .

So the intersection point is $(3, 2)$ .

Watch out

A common mistake is to solve incorrectly or forget to check that the point satisfies both original equations. Always verify: $2(3)-3(2)=0$ ✓ and $4(3)-5(2)=12-10=2$ ✓.

Step 2: Write the family of lines through $(3,2)$

Any line passing through $(3,2)$ can be written as:

y - 2 = m(x - 3)

where $m$ is the slope. Alternatively, in general form:

y - 2 = m(x - 3)

We will use this form and determine $m$ for each condition.

Step 3: Match each condition

(a) Through the point $(2,1)$

Substitute $(2,1)$ into $y - 2 = m(x - 3)$ :

1 - 2 = m(2 - 3) \Rightarrow -1 = m(-1) \Rightarrow m = 1

So the line is $y - 2 = 1(x - 3) \Rightarrow y = x - 1$ , i.e. $x - y - 1 = 0$ .

This matches option (iii) in Column C₂.

(b) Perpendicular to $x + 2y + 1 = 0$

First, find slope of given line: $x + 2y + 1 = 0 \Rightarrow 2y = -x - 1 \Rightarrow y = -\frac{1}{2}x - \frac{1}{2}$ , so slope $m_1 = -\frac{1}{2}$ .

For perpendicular lines, product of slopes = $-1$ :

m \times \left(-\frac{1}{2}\right) = -1 \Rightarrow m = 2

Thus line: $y - 2 = 2(x - 3) \Rightarrow y - 2 = 2x - 6 \Rightarrow 2x - y = 4$ .

This matches option (i).

Tip

Remember: perpendicular slope is the negative reciprocal. If $m_1 = -\frac{1}{2}$ , then $m = 2$ (flip and change sign).

(c) Parallel to $3x - 4y + 5 = 0$

Rewrite: $3x - 4y + 5 = 0 \Rightarrow 4y = 3x + 5 \Rightarrow y = \frac{3}{4}x + \frac{5}{4}$ , so slope $m = \frac{3}{4}$ .

Parallel lines have equal slopes, so $m = \frac{3}{4}$ .

Line: $y - 2 = \frac{3}{4}(x - 3)$ .

Multiply through by 4:

4y - 8 = 3x - 9 \Rightarrow 3x - 4y - 1 = 0

This matches option (iv).

(d) Equally inclined to the axes …

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