Is Transformation of Axes part of the EAPCET Mathematics syllabus?

Yes. Transformation of Axes 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 Transformation of Axes?

Taking h as the whole coefficient of xy; in ax² + 2hxy + by², h is half of it. Substituting x = X − h, y = Y − k when the origin is shifted to (h, k); the correct substitution is x = X + h, y = Y + k. Swapping the rotation formulas; old in terms of new is x = X cos θ − Y sin θ, y = X sin θ + Y cos θ. Expecting the constant term to stay the same under a translation; only the second-degree terms are unchanged.

EAPCET Mathematics · Intermediate first year · Paper IB

Transformation of Axes

Transformation of Axes shows how equations change when the origin is shifted or the axes are rotated, including how to remove the xy-term from a second-degree equation.

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

Key formulas and results

Shifting the origin to $(h, k)$ : $x = X + h$ , $y = Y + k$
Rotating the axes by $\theta$ : $x = X\cos\theta - Y\sin\theta$ , $y = X\sin\theta + Y\cos\theta$
The $xy$ -term of $ax^{2} + 2hxy + by^{2}$ disappears when $\tan 2\theta = \dfrac{2h}{a - b}$
Under rotation, $a + b$ and $ab - h^{2}$ stay the same (invariants)

What this chapter covers

Translation of axes
Rotation of axes
Removing the first-degree and xy terms

How it is usually asked

Find the transformed equation after a shift or rotation
Find the angle of rotation that removes the xy-term
Find a point’s coordinates in the new system

Understanding Transformation of Axes

Transforming the axes changes how a curve is described without moving the curve itself. In a translation the origin moves to a new point (h, k) and the axes stay parallel to the old ones, so the old and new coordinates are related by x = X + h and y = Y + k. Translation is used to remove first-degree terms and bring a conic to a simpler standard form.

In a rotation the origin stays fixed and the axes turn through an angle θ. The old coordinates are then x = X cos θ − Y sin θ and y = X sin θ + Y cos θ. Substituting these into ax² + 2hxy + by² shows that the XY-term vanishes when tan 2θ = 2h/(a − b), and when a = b the required angle is π/4.

Some quantities never change under rotation: a + b and ab − h², and for a translation the second-degree terms remain the same. These invariants let you check a transformed equation in seconds. EAPCET questions ask for the new equation after a shift, the angle that removes the xy-term, or a point’s coordinates in the new system.

Worked example

EAPCET-style question

The angle through which the axes must be rotated to remove the $xy$ -term from $7x^{2} + 2\sqrt{3}xy + 5y^{2} = 16$ is

(1) $\dfrac{\pi}{6}$
(2) $\dfrac{\pi}{3}$
(3) $\dfrac{\pi}{4}$
(4) $\dfrac{\pi}{12}$

Solution

Compare with $ax^{2} + 2hxy + by^{2}$ : $a = 7$ , $b = 5$ and $2h = 2\sqrt{3}$ , so $h = \sqrt{3}$ .
$\tan 2\theta = \dfrac{2h}{a - b} = \dfrac{2\sqrt{3}}{2} = \sqrt{3}$ .
So $2\theta = \tfrac{\pi}{3}$ and $\theta = \tfrac{\pi}{6}$ .
Check with invariants: the new coefficients $A, B$ satisfy $A + B = 12$ and $AB = 35 - 3 = 32$ , giving 8 and 4, so the $xy$ -term is gone.

Answer: $\dfrac{\pi}{6}$

Common mistakes to avoid

Taking h as the whole coefficient of xy; in ax² + 2hxy + by², h is half of it.
Substituting x = X − h, y = Y − k when the origin is shifted to (h, k); the correct substitution is x = X + h, y = Y + k.
Swapping the rotation formulas; old in terms of new is x = X cos θ − Y sin θ, y = X sin θ + Y cos θ.
Expecting the constant term to stay the same under a translation; only the second-degree terms are unchanged.

How to prepare it

Use the invariants a + b and ab − h² to check a transformed equation quickly.

Once the formulas feel familiar, test them the way the exam does: practise Transformation of Axes 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 Transformation of Axes 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 Transformation of Axes part of the EAPCET Mathematics syllabus?: Yes. Transformation of Axes 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 Transformation of Axes for EAPCET?: Use the invariants a + b and ab − h² to check a transformed equation quickly.
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 Transformation of Axes?: Taking h as the whole coefficient of xy; in ax² + 2hxy + by², h is half of it. Substituting x = X − h, y = Y − k when the origin is shifted to (h, k); the correct substitution is x = X + h, y = Y + k. Swapping the rotation formulas; old in terms of new is x = X cos θ − Y sin θ, y = X sin θ + Y cos θ. Expecting the constant term to stay the same under a translation; only the second-degree terms are unchanged.