JEE Mains Maths · Straight Lines

Family of Lines and Locus

Families of lines through a fixed point, three lines that meet at one point, and the locus of a point that moves under a condition, found by eliminating the parameter.

All Straight Lines notes JEE Mains Maths strategy

Why this matters

Ten PYQs, nine of them multiple choice, and one from 2026. Four use a family of lines through a fixed point, or ask when three lines fail to form a triangle. Six find the locus of a moving point by eliminating a slope, an angle or a coordinate. Two ideas cover the page.

Concept 1 of 2: Lines through a fixed point

L_1+\lambda L_2=0

is every line through the meeting point of

L_1=0

and

L_2=0

, except

L_2

itself. An equation that is linear in a parameter can be sorted into this shape, and then the fixed point is where

L_1=L_2=0

. Three lines meet at one point exactly when the determinant of their coefficients is 0. Three lines fail to form a triangle when two of them are parallel or all three meet at one point.

Definition

Through $L_1\cap L_2$ : $L_1+\lambda L_2=0$ .
Fixed point: sort the equation into $L_1+\lambda L_2=0$ , then solve $L_1=L_2=0$ .
Concurrent: $\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}=0$ .
No triangle: two lines parallel, or all three concurrent.
Of the lines through $P$ , the one farthest from $Q$ is perpendicular to $PQ$ , at distance $PQ$ .

Family through a point

L_1+\lambda L_2=0

Worked example

Find the fixed point of the lines

(2+\lambda)x+(1-\lambda)y=3+\lambda

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 8 Apr 2026 Shift 2 · Q60Moderate

Example 1 · Straight Lines · Family of Lines and Locus

If a straight line drawn through the point of intersection of the lines

4x+ 3y- 1 = 0

and

3x+ 4y- 1 = 0

, meet the co-ordinate axes at the points P and Q , then the locus of the mid point of PQ is:

Count every way to fail

Three lines do not form a triangle when any two are parallel or when all three meet at one point. Each case gives its own values of the parameter, and the answer needs all of them.

Drill 3 more on lines through a fixed point

Concept 2 of 2: Locus of a moving point

Call the moving point

(h,k)

. Write

h

and

k

in terms of whatever moves it: a slope, an angle or another point's coordinate. Then eliminate that parameter and rename

h,k

x,y

. For angles,

\cos^2\theta+\sin^2\theta=1

or the tan addition formula usually does the elimination.

Definition

Set the point as $(h,k)$ and express both in the parameter.
Eliminate the parameter; then write $x,y$ for $h,k$ .
The midpoint of the intercepts of $\frac xa+\frac yb=1$ is $\left(\frac a2,\frac b2\right)$ .
A fixed area on a fixed base: the locus is a line parallel to the base.

Elimination

h=f(t),\ k=g(t)\ \Rightarrow\ F(h,k)=0

Worked example

A line through

(2,3)

meets the axes at

A

and

B

. Find the locus of the midpoint of

AB

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 7 Apr 2025 · Q58Moderate

Example 2 · Straight Lines · Family of Lines and Locus

If for

\theta\in\left\lbrack -\frac{\pi}{3},0 \right\rbrack

, the points

(x,y) =\left( 3\tan\left( \theta+\frac{\pi}{3} \right),2\tan\left( \theta+\frac{\pi}{6} \right) \right)

lie on

xy +\alpha x +\beta y +\gamma= 0

, then

\alpha^{2}+\beta^{2}+\gamma^{2}

is equal to :

Keep the restrictions

Eliminating the parameter can add points the moving point never reaches. If

\theta

or a coordinate is restricted, the locus is only the matching part of the curve, and the options may differ only there.

Drill 5 more on locus of a moving point

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)

Lines through a fixed point
Family through a point
$L_1+\lambda L_2=0$
Locus of a moving point
Elimination
$h=f(t),\ k=g(t)\ \Rightarrow\ F(h,k)=0$

Watch out for (2)

Count every way to fail
→ Lines through a fixed point
Keep the restrictions
→ Locus of a moving point

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.

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