JEE Mains Maths · Straight Lines

Image of a Point and Reflected Rays

The image of a point in a line, found from the foot of the perpendicular, and its use for reflected rays of light and for shortest paths that touch a line.

All Straight Lines notes JEE Mains Maths strategy

Why this matters

Thirteen PYQs, eight of them multiple choice, and two from 2026. Seven reflect a point, a centroid or a whole triangle in a line, one of them followed by a translation and a rotation. Six follow a ray of light reflected in a mirror line, or the shortest path that touches a line. Two ideas cover the page.

Concept 1 of 2: Image of a point in a line

The image of

P

in a line is as far behind the line as

P

is in front of it, along the normal. So move

P

along

(a,b)

by twice the step that reaches the foot of the perpendicular. The line is then the perpendicular bisector of

P

and its image. Reflection keeps lengths and angles, so the image of a triangle's centroid is the centroid of its image, and a circle's image has the reflected centre and the same radius.

Definition

Image: $\frac{x-x_1}a=\frac{y-y_1}b=-\frac{2(ax_1+by_1+c)}{a^2+b^2}$ .
Foot: the same with 2 replaced by 1.
In $y=x$ : $(x,y)\to(y,x)$ ; in the x-axis: $(x,y)\to(x,-y)$ .
The mirror is the perpendicular bisector of $P$ and its image.

Image in a line

\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{2(ax_1+by_1+c)}{a^2+b^2}

Worked example

Find the image of

(4,1)

2x+y=4

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 21 Jan 2026 Shift 2 · Q75Moderate

Example 1 · Straight Lines · Image of a Point and Reflected Rays

If P is a point on the circle

x^{2} + y^{2} = 4,Q

is a point on the straight line

5x + y + 2 = 0

and

x - y + 1 = 0

is the perpendicular bisector of PQ , then 13 times the sum of abscissa of all such point P is

\_\_\_\_

Twice, not once

The formula with 1 gives the foot of the perpendicular; with 2 it gives the image. Check the answer: the midpoint of

P

and its image must lie on the line.

Drill 6 more on image of a point in a line

Concept 2 of 2: Reflected rays and shortest paths

A ray reflected in a mirror line looks as if it comes from the image of its source. So the reflected ray is the line through the image of the source and any point the reflected ray passes through. The same idea gives the shortest path from

A

to a line and on to

B

, with

A

and

B

on the same side: reflect

A

, join the image to

B

, and the crossing point is where the path touches the line.

Definition

Reflected ray: through the image $A'$ of the source $A$ and the point it reaches.
Incident ray: through the source and the image of the point the reflected ray reaches.
Shortest path $A\to$ line $\to B$ : length $A'B$ , turning where $A'B$ meets the line.
The angle of incidence equals the angle of reflection.

Path via a mirror

AR+RB\ge A'B,\ \text{equality when } R \text{ is on } A'B

Worked example

A ray from

(1,3)

reflects off the x-axis and passes through

(5,1)

. Where does it hit the axis?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 5 Apr 2026 Shift 2 · Q72Moderate

Example 2 · Straight Lines · Image of a Point and Reflected Rays

From the point

( - 1, - 1)

, two rays are sent making angles of

45^{\circ}

with the line

x + y = 0

. These rays get reflected from the mirror

x + 2y = 1

. If the equations of the reflected rays are

ax + by = 9

and

cx + dy = 7,a,b,c,d \in Z

, then the value of

ad + bc

\_\_\_\_

Reflect the right point

The reflected ray passes through the image of the source. Reflecting the far point instead gives the incident ray. Check which ray the question asks for before you reflect.

Drill 5 more on reflected rays and shortest paths

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Image of a point in a line
Image in a line
$\frac{x-x_1}{a}=\frac{y-y_1}{b}=-\frac{2(ax_1+by_1+c)}{a^2+b^2}$
Reflected rays and shortest paths
Path via a mirror
$AR+RB\ge A'B,\ \text{equality when } R \text{ is on } A'B$

Watch out for (2)

Twice, not once
→ Image of a point in a line
Reflect the right point
→ Reflected rays and shortest paths

Test yourself on Straight Lines

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Straight Lines chapter test Drill all 13 questions