Question 1

The slope of the line joining $(2,3)$ and $(4,7)$ is:

Accepted Answer

**Answer: (a) $2$.** $m=\dfrac{7-3}{4-2}=\dfrac{4}{2}=2.$

Question 2

The slope of the line $2x+3y=6$ is:

Accepted Answer

**Answer: (a) $-	frac23$.** $3y=-2x+6\Rightarrow y=-\dfrac23x+2$, so the slope is $-\dfrac23.$

Question 3

The equation of the line through $(0,4)$ parallel to $3x-y=5$ is:

Accepted Answer

**Answer: (a) $y=3x+4$.** $3x-y=5\Rightarrow y=3x-5$, slope $3$. A parallel line has slope $3$ and $y$-intercept $4$: $y=3x+4.$

Question 4

The slope of a line perpendicular to the line joining $(1,2)$ and $(3,8)$ is:

Accepted Answer

**Answer: (a) $-	frac13$.** Slope of the join $=\dfrac{8-2}{3-1}=3$. Perpendicular slope $=-\dfrac1{3}.$

Question 5

**Assertion (A):** The lines $y=2x+3$ and $2x-y+7=0$ are parallel. **Reason (R):** Two lines are parallel when their slopes are equal.

Accepted Answer

**Answer: (a)** The first line has slope $2$; the second is $y=2x+7$, slope $2$. Equal slopes mean the lines are parallel, so R correctly explains A.

Question 6

Find the slope and the $y$-intercept of the line $4x-2y+6=0$.

Accepted Answer

$2y=4x+6\Rightarrow y=2x+3.$ Slope $=2$ and $y$-intercept $=3.$

Question 7

Find the equation of the line passing through $(2,-3)$ with slope $-4$.

Accepted Answer

$y-y_1=m(x-x_1)$: $y-(-3)=-4(x-2)\Rightarrow y+3=-4x+8\Rightarrow 4x+y-5=0.$

Question 8

Find the equation of the line passing through the points $(3,4)$ and $(-1,2)$.

Accepted Answer

Slope $m=\dfrac{2-4}{-1-3}=\dfrac{-2}{-4}=\dfrac12.$ $y-4=\dfrac12(x-3)\Rightarrow 2y-8=x-3\Rightarrow x-2y+5=0.$

Question 9

Find the equation of the line passing through $(-2,3)$ and perpendicular to the line $2x-3y=6$.

Accepted Answer

$2x-3y=6\Rightarrow y=\dfrac23x-2$, slope $\dfrac23$. Perpendicular slope $=-\dfrac32.$ $y-3=-\dfrac32(x+2)\Rightarrow 2y-6=-3x-6\Rightarrow 3x+2y=0.$

Question 10

A line passes through the point $(4,3)$ and makes equal intercepts on the two axes. Find its equation.

Accepted Answer

Equal intercepts mean the line is $\dfrac{x}{a}+\dfrac{y}{a}=1$, i.e. $x+y=a.$ It passes through $(4,3)$: $4+3=a\Rightarrow a=7.$ The equation is $x+y=7.$

Question 11

Points $A(1,4)$ and $B(3,0)$ are given. (i) Find the equation of line $AB$. (ii) Find the equation of the perpendicular bisector of $AB$. (iii) Verify whether the point $(0,1)$ lies on this perpendicular bisector.

Accepted Answer

(i) Slope of $AB=\dfrac{0-4}{3-1}=-2.$ Using $A(1,4)$ : $y-4=-2(x-1)\Rightarrow 2x+y=6.$

(ii) Mid-point of $AB=\left(\dfrac{1+3}{2},\dfrac{4+0}{2}\right)=(2,2).$ Perpendicular slope $=\dfrac12.$

$y-2=\dfrac12(x-2)\Rightarrow 2y-4=x-2\Rightarrow x-2y+2=0.$

(iii) Put $(0,1)$ : $0-2(1)+2=0$ , which holds, so $(0,1)$ lies on the perpendicular bisector.

Question 12

A line $L$ passes through the point $P(-2,3)$ and is parallel to the line joining $A(2,-1)$ and $B(6,3)$. (i) Find the equation of $L$. (ii) Find the coordinates of the points where $L$ cuts the $x$-axis and the $y$-axis. (iii) Find the point where $L$ meets the line $2x+y=4$.

Accepted Answer

(i) Slope of $AB=\dfrac{3-(-1)}{6-2}=\dfrac{4}{4}=1.$ Line $L$ has slope $1$ through $P(-2,3)$ : $y-3=1(x+2)\Rightarrow y=x+5.$

(ii) $x$ -axis ( $y=0$ ): $0=x+5\Rightarrow x=-5$ , point $(-5,0).$ $y$ -axis ( $x=0$ ): $y=5$ , point $(0,5).$

(iii) Solve $y=x+5$ and $2x+y=4$ : $2x+(x+5)=4\Rightarrow 3x=-1\Rightarrow x=-\dfrac13,\ y=x+5=\dfrac{14}{3}.$ The point is $\left(-\dfrac13,\dfrac{14}{3}\right).$

Question 13

A straight stretch of road on a map passes through the points $A(0,2)$ and $B(4,10)$ .

(i) Find the slope of the road.

(ii) Find the equation of the road.

(iii) Does the point $(3,8)$ lie on this road?

(iv) Find the equation of a road parallel to it that passes through $(0,-1)$ .

Accepted Answer

(i) Slope $=\dfrac{10-2}{4-0}=\dfrac{8}{4}=2.$

(ii) The $y$ -intercept is $2$ , so $y=2x+2.$

(iii) At $x=3$ : $y=2(3)+2=8$ , so $(3,8)$ does lie on the road.

(iv) A parallel road has slope $2$ ; through $(0,-1)$ its equation is $y=2x-1.$

Question 14

Are these Equation of a Line important questions free?

Accepted Answer

Yes. All 13 ICSE Class 10 Maths important questions for Equation of a Line are free, with full model answers and no login required.

Question 15

Do these Equation of a Line questions follow the latest ICSE syllabus?

Accepted Answer

Yes — they are aligned to the CISCE latest syllabus syllabus for ICSE Class 10 Maths, so nothing here is outside the current course.

Question 16

How should I practise the Equation of a Line important questions?

Accepted Answer

Attempt each question on paper first, then reveal the model answer to check your method — not just the final result. Re-do anything you got wrong the same day.

Question 17

What types of questions are covered for Equation of a Line?

Accepted Answer

A full mix — multiple-choice questions, assertion–reason questions, very short answer questions, short answer questions, long answer questions, case-based questions — so every format in the ICSE paper is covered.

Question type	Count	Marks
MCQ	4	1
Assertion–Reason	1	1
Very Short	2	2
Short Answer	3	3
Long Answer	2	5
Case-based	1	4

Equation of a Line — Important Questions

About Equation of a Line

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Important questions with answers

Multiple-choice questions (1 mark)

Assertion–Reason questions (1 mark)

Very short answer questions (2 marks)

Short answer questions (3 marks)

Long answer questions (5 marks)

Case-based questions (4 marks)

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