NDA Maths · Lines

Equations, Slope & Family of Lines

The slope of a line and the standard forms of its equation, the intercept form, the family of lines through a point or an intersection, and reflections in a line.

All Lines notes NDA Maths strategy

Why this matters

Every other tool in the chapter starts from a line's equation. Knowing which form to reach for — slope-intercept, point-slope, intercept, or the family L₁+λL₂ — turns most questions into one substitution.

Concept 1 of 4

Slope and the forms of a line

Intuition

A line is fixed by a point and a direction (its slope). The slope is the tangent of the angle it makes with the x-axis. Pick the form that matches what you're given — a point and slope, two points, or intercepts.

Definition

Slope $m=\tan\theta=\dfrac{y_2-y_1}{x_2-x_1}$ . Forms:

Slope-intercept: $y=mx+c$ . Point-slope: $y-y_1=m(x-x_1)$ .
Two-point: $\dfrac{y-y_1}{x-x_1}=\dfrac{y_2-y_1}{x_2-x_1}$ .
General: $ax+by+c=0$ has slope $-a/b$ . Normal form: $x\cos\theta+y\sin\theta=p$ ( $p$ = distance from origin).

Worked example

Find the equation of the line through

(2,3)

with slope

4

Point-slope: $y-3=4(x-2)$ .
Simplify: $y=4x-5$ .

Answer:

4x-y-5=0

Practice this conceptself-check · 4 quick reps

Try it yourself

What is the slope of

3x-4y+7=0

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Slope of $ax+by+c=0$ ?
2.
Slope in terms of the angle $\theta$ with x-axis?
3.
Point-slope form?
4.
Normal form of a line?

Drill 9 more on slope and the forms of a line

Concept 2 of 4

Intercept form and intercepts

Intuition

When a line's x- and y-intercepts matter, the intercept form

\tfrac{x}{a}+\tfrac{y}{b}=1

reads them off directly. Many questions give a relation between the intercepts (their sum, or a midpoint) and ask for the line.

Definition

Intercept form: $\dfrac{x}{a}+\dfrac{y}{b}=1$ , where $a$ is the x-intercept and $b$ the y-intercept. From $ax+by+c=0$ : x-intercept $=-c/a$ , y-intercept $=-c/b$ . If $(h,k)$ is the midpoint of the intercept segment, then $a=2h$ , $b=2k$ .

Worked example

A line has x-intercept

4

and y-intercept

2

. Find its equation and the sum of intercepts.

Intercept form: $\dfrac{x}{4}+\dfrac{y}{2}=1\Rightarrow x+2y=4$ .
Sum of intercepts $=4+2=6$ .

Answer:

x+2y=4

; sum

=6

Practice this conceptself-check · 4 quick reps

Try it yourself

(3,4)

is the midpoint of the segment a line cuts between the axes, find the line.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Intercept form of a line?
2.
x-intercept of $ax+by+c=0$ ?
3.
Midpoint of intercepts is $(h,k)$ : intercepts?
4.
Sum of intercepts of $\tfrac{x}{4}+\tfrac{y}{2}=1$ ?

From the bank · past-year question

Example 2LinesMODERATE

What is the sum of the intercepts of the line

\dfrac{x}{a^2}+\dfrac{y}{b^2}=\dfrac{2}{a^2+b^2}

on the coordinate axes?

[Q60 · Sep · 2025]

Drill 5 more on intercept form and intercepts

Concept 3 of 4

Family of lines and concurrency

Intuition

Any line through the intersection of

L_1=0

and

L_2=0

can be written

L_1+\lambda L_2=0

— without ever finding the intersection point. Choose

\lambda

from one extra condition. Three lines are concurrent when their intersection is shared.

Definition

Family (pencil): through $L_1\cap L_2$ , every line is $L_1+\lambda L_2=0$ ; fix $\lambda$ from a point or a slope condition. Parallel/perpendicular through a point: keep the same (or negative-reciprocal) slope. Concurrency: three lines are concurrent iff $\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}=0$ . If coefficients $A,B,C$ are in AP, $Ax+2By+C=0$ passes through the fixed point $(1,-2)$ .

Worked example

Find the line through the intersection of

x+y-1=0

and

2x-y-2=0

that passes through

(1,2)

Family: $(x+y-1)+\lambda(2x-y-2)=0$ .
At $(1,2)$ : $(1+2-1)+\lambda(2-2-2)=0\Rightarrow 2-2\lambda=0\Rightarrow\lambda=1$ .
Sum: $3x-3=0\Rightarrow x=1$ .

Answer:

x=1

Practice this conceptself-check · 4 quick reps

Try it yourself

For what

k

(k-3)x-y+2=0

parallel to

x+y=1

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Family of lines through $L_1\cap L_2$ ?
2.
Concurrency condition for 3 lines?
3.
Parallel lines have slopes that are?
4.
$Ax+2By+C=0$ with $A,B,C$ in AP passes through?

From the bank · past-year question

Example 3LinesMODERATE

If the lines

3y+4x=1

y=x+5

and

5y+bx=3

are concurrent, then what is the value of b?

[Q57 · Apr · 2019]

Drill 8 more on family of lines and concurrency

Concept 4 of 4

Image of a point and reflections

Intuition

The image of a point in a line is its mirror reflection: the line is the perpendicular bisector of the segment joining the point and its image. Use 'midpoint lies on the line' plus 'segment ⟂ line' to find the image, or recover the mirror from a point–image pair.

Definition

If $P'$ is the image of $P$ in line $L$ : the midpoint of $PP'$ lies on $L$ , and $PP'\perp L$ . These two conditions pin down $P'$ (or the mirror line). Foot of perpendicular from $P$ to $L$ is the midpoint of $PP'$ .

Worked example

Find the image of

(1,2)

in the line

y=x

Reflection in $y=x$ swaps coordinates.
Image $=(2,1)$ .

Answer:

(2,1)

Practice this conceptself-check · 4 quick reps

Try it yourself

The image of

(-4,2)

in a line is

(4,-2)

. What line is the mirror?

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
A line is the ___ of a point and its image.
2.
Image of $(a,b)$ in $y=x$ ?
3.
Image of $(a,b)$ in the x-axis?
4.
Foot of perpendicular is the ___ of P and its image.

From the bank · past-year question

Example 4LinesMODERATE

If the image of the point

(-4, 2)

by a line mirror is

(4, -2)

, then what is the equation of the line mirror?

[Q60 · Apr · 2021]

Drill 2 more on image of a point and reflections

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1LinesEASY

Under what condition will the lines

m^2 x + ny - 1 = 0

and

n^2 x - my + 2 = 0

be perpendicular?

[Q51 · Apr · 2025]

Example 2LinesEASY

The equation of a straight line which cuts off an intercept of 5 units on negative direction of

y

-axis and makes an angle 120° with positive direction of

x

-axis is

[Q55 · Sep · 2017]

Example 3LinesMODERATE

The equation of the line passing through the point

(2, 3)

and the point of intersection of lines

2x - 3y + 7 = 0

and

7x + 4y + 2 = 0

[Q56 · Sep · 2017]

Example 4LinesMODERATE

ABC is an equilateral triangle and AD is the altitude on BC. If the coordinates of A are

(1, 2)

and that of D are

(-2, 6)

, then what is the equation of BC?

[Q55 · Apr · 2025]

Example 5LinesEASY

The equation

ax + by + c = 0

represents a straight line

[Q51 · Sep · 2019]

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