Question 303 marksShort AnswerSection C

If x=h+acos⁡θ,y=k+bsin⁡θx = h + a\cos\theta, y = k + b\sin\theta, then prove that :

(x−ha)2+(y−kb)2=1\left(\frac{x-h}{a}\right)^2 + \left(\frac{y-k}{b}\right)^2 = 1

Introduction to Trigonometry
Trigonometric Identities (Parametric Form of an Ellipse)

Official answer

Proved: (x−ha)2+(y−kb)2=1\left(\frac{x-h}{a}\right)^2 + \left(\frac{y-k}{b}\right)^2 = 1. From x=h+acos⁡θx = h + a\cos\theta, x−ha=cos⁡θ\frac{x-h}{a} = \cos\theta; from y=k+bsin⁡θy = k + b\sin\theta, y−kb=sin⁡θ\frac{y-k}{b} = \sin\theta. Squaring both, (x−ha)2=cos⁡2θ\left(\frac{x-h}{a}\right)^2 = \cos^2\theta and (y−kb)2=sin⁡2θ\left(\frac{y-k}{b}\right)^2 = \sin^2\theta. Adding, (x−ha)2+(y−kb)2=cos⁡2θ+sin⁡2θ\left(\frac{x-h}{a}\right)^2 + \left(\frac{y-k}{b}\right)^2 = \cos^2\theta + \sin^2\theta. Since sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 for all θ, the sum equals 1, proving the required identity holds for every value of θ.

Answer keywords:parametric equationstrigonometric identitysin squared plus cos squared equals oneeliminate parameter thetaellipse parametric formalgebraic proof

Marking scheme

  1. 1.1 mark: correctly isolating x−ha=cos⁡θ\frac{x-h}{a} = \cos\theta and y−kb=sin⁡θ\frac{y-k}{b} = \sin\theta from the given equations.
  2. 2.1 mark: correctly squaring both expressions to get (x−ha)2=cos⁡2θ\left(\frac{x-h}{a}\right)^2 = \cos^2\theta and (y−kb)2=sin⁡2θ\left(\frac{y-k}{b}\right)^2 = \sin^2\theta.
  3. 3.1 mark: correctly adding the two squared expressions and applying the identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 to reach the required result.

Hint

Isolate cos θ from the x-equation and sin θ from the y-equation, square both, and add using sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.

Quick oral answer

I isolate cos θ and sin θ from the two given equations, square each, and add them; since sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 always, the sum equals 1, which proves the required identity.

Common mistakes

  1. 1.Substituting a specific value of θ (like 0° or 90°) to 'verify' rather than proving the identity algebraically for a general θ.
  2. 2.Forgetting to divide by a and b respectively while isolating cos θ and sin θ, leading to an incorrect squared expression.
  3. 3.Sign or arrangement errors when subtracting h and k from x and y respectively before isolating the trigonometric ratios.

Analysis and explanation

Eliminate θ from the two parametric equations using the identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1.


Concept

  • Isolate cos θ and sin θ from the given equations, then square and add.
  • These equations are the parametric form of an ellipse centred at (h, k) with semi-axes a, b.

Key Points

  • x−ha=cos⁡θ\frac{x-h}{a} = \cos\theta and y−kb=sin⁡θ\frac{y-k}{b} = \sin\theta; squaring and adding gives cos⁡2θ+sin⁡2θ=1\cos^2\theta + \sin^2\theta = 1.

Common Mistakes

  • Substituting a specific numeric value of θ instead of proving the identity holds for all θ.
  • Sign/division errors while isolating cos θ or sin θ.

Real-world Application

  • This parametrisation links trigonometry to conics (circle when a=ba = b), used in describing elliptical orbits and elliptical arches/gears in engineering.

Interesting facts

The given parametric equations x=h+acos⁡θ,y=k+bsin⁡θx = h + a\cos\theta, y = k + b\sin\theta actually represent an ellipse centred at (h, k) with semi-major/minor axes a and b — when a=ba = b, this reduces to the parametric form of a circle.

Parametric equations, using a third variable (parameter) like θ to define x and y, are widely used in physics and engineering to describe motion along curved paths, such as projectile motion or planetary orbits.

The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 used here is one of the three fundamental trigonometric identities and is essentially a restatement of the Pythagoras theorem for a right triangle inscribed in a unit circle.

Question details

Q30 of 56 questions

Board
CBSE
Class
Class 10
Subject
Mathematics (Standard)
Year
2026
Section
Section C
Marks
3
Type
Short Answer
Chapter
Introduction to Trigonometry
Topic
Trigonometric Identities (Parametric Form of an Ellipse)
Bloom's level
analyze
Syllabus topic
Unit V: Trigonometry — Chapter: Introduction to Trigonometry (Application of sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 to eliminate a parameter)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

What geometric curve do these parametric equations represent?

They represent an ellipse centred at (h, k) with semi-axis lengths a (along x) and b (along y); if a = b, the curve becomes a circle of radius a centred at (h, k).

Can this proof be done by substituting a particular angle for θ?

No — since the question says 'prove that', the identity must hold for every value of θ, so it must be shown algebraically using sin⁡2θ+cos⁡2θ=1\sin^2\theta+\cos^2\theta=1, not verified for one specific angle only.