NDA Maths · Lines

Distance, Section & Locus

Distances between points, from a point to a line, and between parallel lines; the section formula for dividing a segment; and locus equations from a geometric condition.

All Lines notes NDA Maths strategy

Why this matters

These three formulas underpin half the chapter. The point-to-line distance and the section formula appear constantly, and 'locus' questions are just a geometric condition translated into an equation.

Concept 1 of 3

Distance: point-point, point-line, parallel lines

Intuition

Three distances recur: between two points (Pythagoras), from a point to a line (perpendicular distance using the normalised equation), and between two parallel lines (difference of constants over the same normaliser).

Definition

Two points: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ .
Point to line: distance from $(x_0,y_0)$ to $ax+by+c=0$ is $\dfrac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$ .
Parallel lines $ax+by+c_1=0$ , $ax+by+c_2=0$ : $\dfrac{|c_1-c_2|}{\sqrt{a^2+b^2}}$ (make the $a,b$ match first).

Worked example

Find the distance from

(1,2)

to the line

3x+4y-10=0

$\dfrac{|3(1)+4(2)-10|}{\sqrt{3^2+4^2}}=\dfrac{|1|}{5}$ .

Answer:

\tfrac15

Practice this conceptself-check · 4 quick reps

Try it yourself

Find the distance between

3x+4y=9

and

3x+4y=4

Practice — Level 1 (4 reps)

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

1.
Distance from $(x_0,y_0)$ to $ax+by+c=0$ ?
2.
Distance between $ax+by+c_1=0,\ ax+by+c_2=0$ ?
3.
Distance from $(1,2)$ to $3x+4y-10=0$ ?
4.
Before using the parallel-line formula, ensure?

From the bank · past-year question

Example 1LinesEASY

What is the distance between

3x+4y=9

and

6x+8y=15

[Q64 · Apr · 2018]

Drill 7 more on distance: point-point, point-line, parallel lines

Concept 2 of 3

Section formula and ratios

Intuition

The point dividing a segment in a given ratio is a weighted average of the endpoints. Run it backwards to find the ratio in which a point (or a line) divides a segment, or forwards to find midpoints and divisions.

Definition

Point dividing $P(x_1,y_1)$ , $Q(x_2,y_2)$ internally in ratio $m:n$ : $\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$ . External division uses $m:-n$ . Midpoint is the $1:1$ case. Collinearity of $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ : area $=0$ (equivalently equal slopes).

Worked example

Find the point dividing

(1,2)

and

(4,8)

internally in ratio

2:1

$x=\dfrac{2(4)+1(1)}{3}=3$ , $y=\dfrac{2(8)+1(2)}{3}=6$ .

Answer:

(3,6)

Practice this conceptself-check · 4 quick reps

Try it yourself

In what ratio does

x+y=4

divide the segment from

P(1,1)

Q(5,7)

Practice — Level 1 (4 reps)

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

1.
Internal division $m:n$ x-coordinate?
2.
Midpoint is which ratio?
3.
External division uses ratio?
4.
Three points collinear ⇒ area?

From the bank · past-year question

Example 2LinesMODERATE

What is the ratio in which the point

C\!\left(-\dfrac{2}{7}, -\dfrac{20}{7}\right)

divides the line joining the points

A(-2, -2)

and

B(2, -4)

[Q56 · Apr · 2017]

Drill 8 more on section formula and ratios

Concept 3 of 3

Locus from a condition

Intuition

A locus is the set of points satisfying a rule. Let the moving point be

(x,y)

, write the geometric condition algebraically, and simplify — the equation that survives is the locus. 'Equidistant from two points' always gives the perpendicular bisector.

Definition

Set $P=(x,y)$ , translate the condition (equidistant, fixed ratio, sum/difference of distances), and reduce. **Equidistant from $A,B$ :** $PA^2=PB^2$ ⇒ the perpendicular bisector of $AB$ (a line). Equidistant from two lines ⇒ the angle bisectors.

Worked example

Find the locus of points equidistant from

A(1,2)

and

B(3,4)

$(x-1)^2+(y-2)^2=(x-3)^2+(y-4)^2$ .
Expand & cancel: $-2x-4y+5=-6x-8y+25\Rightarrow 4x+4y=20\Rightarrow x+y=5$ .

Answer:

x+y=5

(the perpendicular bisector).

Practice this conceptself-check · 4 quick reps

Try it yourself

Locus of points equidistant from

(2a,0)

and

(0,3a)

Practice — Level 1 (4 reps)

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

1.
Locus equidistant from two points is?
2.
First step in a locus problem?
3.
Locus equidistant from two lines?
4.
Condition for $P$ equidistant from $A,B$ ?

From the bank · past-year question

Example 3LinesEASY

What is the equation of the locus of the mid-point of the line segment obtained by cutting the line

x+y=p

(where

p

is a real number) by the coordinate axes?

[Q89 · Apr · 2022]

Drill 4 more on locus from a condition

Mastery check — 5 interleaved questions

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

Example 1LinesMODERATE

The distance of the point

(1, 3)

from the line

2x + 3y = 6

, measured parallel to the line

4x + y = 4

, is

[Q50 · Sep · 2017]

Example 2LinesHARD

Two points

P

and

Q

lie on line

y=2x+3

. These two points

P

and

Q

are at a distance of

\sqrt{2}

units from another point

R(1,5)

. What are the coordinates of

P

and

Q

[Q60 · Apr · 2024]

Example 3LinesMODERATE

The points

(1,3)

and

(5,1)

are two opposite vertices of a rectangle. The other two vertices lie on the line

y=2x+c

. What is the value of c?

[Q56 · Apr · 2019]

Example 4LinesMODERATE

Consider the following statements: 1. For an equation of a line

x\cos\theta+y\sin\theta=p

in normal form, the length of the perpendicular from the point

(\alpha,\beta)

to the line is

|\alpha\cos\theta+\beta\sin\theta+p|

. 2. The length of the perpendicular from the point

(\alpha,\beta)

to the line

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

\left|\frac{a\alpha+b\beta-ab}{\sqrt{a^2+b^2}}\right|

. Which of the above statements is/are correct?

[Q51 · Apr · 2019]

Example 5LinesHARD

If the points with coordinates

(-5,0)

(5p^2,10p)

and

(5q^2,10q)

are collinear, then what is the value of

pq

where

p\neq q

[Q81 · Apr · 2022]

Drill every past-year question on this subtopic

22 questions from the bank — paginated, with cart and Word-export support.

Drill the 22 questions

Drill every past-year question on this subtopic

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