Is Pair of Straight lines part of the EAPCET Mathematics syllabus?

Yes. Pair of Straight lines is taught in Intermediate first year · Paper IB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.

How many Mathematics questions are there in EAPCET?

80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.

What mistakes do students make in Pair of Straight lines?

Taking h equal to the full xy coefficient; in ax² + 2hxy + by², h is half of it. Swapping a and b in the slope relations; the sum of slopes is −2h/b and the product is a/b. Testing perpendicularity with h = 0; the lines are perpendicular when a + b = 0, the coefficients of x² and y² adding to zero. In homogenisation, multiplying the first-degree terms by (lx + my)² or the constant by (lx + my); first-degree terms need one factor and the constant needs the square.

EAPCET Mathematics · Intermediate first year · Paper IB

Pair of Straight lines

Pair of Straight Lines studies second-degree equations that represent two lines, the angle between them, their bisectors and the technique of homogenisation.

Subject: Mathematics
Taught in: Intermediate first year · Paper IB
Mathematics in the paper: 80 of 160 questions

Key formulas and results

$ax^{2} + 2hxy + by^{2} = 0$ represents two lines through the origin; sum of slopes $= -\dfrac{2h}{b}$ , product $= \dfrac{a}{b}$
Angle between them: $\tan\theta = \dfrac{2\sqrt{h^{2} - ab}}{|a + b|}$
The lines are perpendicular if $a + b = 0$ and coincident if $h^{2} = ab$
$ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$ is a pair of lines iff $abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$
Bisectors of the angles: $\dfrac{x^{2} - y^{2}}{a - b} = \dfrac{xy}{h}$

What this chapter covers

Homogeneous equation of second degree
Angle between the pair of lines
Angle bisectors
Condition for a general second-degree equation to represent a pair of lines
Homogenisation

How it is usually asked

Find the angle between the lines of a pair
Find the condition for an equation to represent a pair of lines
Use homogenisation to find lines joining the origin to the intersection of a line and a curve

Understanding Pair of Straight lines

A homogeneous equation ax² + 2hxy + by² = 0 represents two straight lines through the origin, because it factorises into two linear factors. Dividing by x² turns it into a quadratic in the slope m = y/x, so the two slopes satisfy m₁ + m₂ = −2h/b and m₁m₂ = a/b. These two relations are the source of almost every formula in the chapter.

From them, the angle between the lines satisfies tan θ = 2√(h² − ab)/|a + b|. The lines are real and distinct when h² > ab, coincident when h² = ab and perpendicular when a + b = 0. The pair of angle bisectors is (x² − y²)/(a − b) = xy/h. The general second-degree equation represents a pair of lines when abc + 2fgh − af² − bg² − ch² = 0.

Homogenisation finds the pair of lines joining the origin to the points where a line meets a curve: rewrite the line as lx + my = 1, then multiply each term of the curve’s equation by the right power of (lx + my) so every term becomes second degree. EAPCET questions focus on angles between the lines, conditions on constants, and homogenisation setups.

Worked example

EAPCET-style question

The acute angle between the lines represented by $2x^{2} + 5xy + 2y^{2} = 0$ is

(1) $\tan^{-1}\tfrac{3}{4}$
(2) $\tan^{-1}\tfrac{4}{3}$
(3) $\tan^{-1}\tfrac{3}{2}$
(4) $\dfrac{\pi}{4}$

Solution

Here $a = 2$ , $b = 2$ and $2h = 5$ , so $h = \tfrac{5}{2}$ .
$h^{2} - ab = \tfrac{25}{4} - 4 = \tfrac{9}{4}$ , so $\sqrt{h^{2} - ab} = \tfrac{3}{2}$ .
$\tan\theta = \dfrac{2 \cdot \frac{3}{2}}{|2 + 2|} = \dfrac{3}{4}$ .
Check: $2x^{2} + 5xy + 2y^{2} = (2x + y)(x + 2y)$ , slopes $-2$ and $-\tfrac{1}{2}$ , and $\left|\dfrac{-2 + \frac{1}{2}}{1 + 1}\right| = \dfrac{3}{4}$ .

Answer: $\tan^{-1}\tfrac{3}{4}$

Common mistakes to avoid

Taking h equal to the full xy coefficient; in ax² + 2hxy + by², h is half of it.
Swapping a and b in the slope relations; the sum of slopes is −2h/b and the product is a/b.
Testing perpendicularity with h = 0; the lines are perpendicular when a + b = 0, the coefficients of x² and y² adding to zero.
In homogenisation, multiplying the first-degree terms by (lx + my)² or the constant by (lx + my); first-degree terms need one factor and the constant needs the square.

How to prepare it

Homogenisation questions repeat in pattern — practise until you can set them up without thinking.

Once the formulas feel familiar, test them the way the exam does: practise Pair of Straight lines questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Pair of Straight lines chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Pair of Straight lines part of the EAPCET Mathematics syllabus?: Yes. Pair of Straight lines is taught in Intermediate first year · Paper IB Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Pair of Straight lines for EAPCET?: Homogenisation questions repeat in pattern — practise until you can set them up without thinking.
How many Mathematics questions are there in EAPCET?: 80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Pair of Straight lines?: Taking h equal to the full xy coefficient; in ax² + 2hxy + by², h is half of it. Swapping a and b in the slope relations; the sum of slopes is −2h/b and the product is a/b. Testing perpendicularity with h = 0; the lines are perpendicular when a + b = 0, the coefficients of x² and y² adding to zero. In homogenisation, multiplying the first-degree terms by (lx + my)² or the constant by (lx + my); first-degree terms need one factor and the constant needs the square.