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Mathematics · Class 11 Science

Ch 16Transformation of Axes — Class 11 Mathematics, concept-first.

Translation of axes means shifting the origin to a new point while keeping the direction of both axes unchanged — the new -axis is parallel to the old -axis, and the new -axis is parallel to the old -axis. Nothing about the curve or the point in the plane changes; only the frame we use to name its coordinates changes.

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Key concepts

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Translation of Axes

Translation of axes is the simplest of the two basic coordinate transformations: the origin is moved to a new point , but the two axes keep their original directions — the new -axis remains parallel to the old -axis and…

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Chapter contents

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2.1

Translation of Axes

Translation of axes means shifting the origin to a new point while keeping the direction of both axes unchanged — the new -axis is parallel to the old -axis, and the new -axis is parallel to the old -…

2.2

Rotation of Axes

Rotation of axes keeps the origin fixed but turns the pair of perpendicular axes through a fixed angle , measured counter-clockwise from the old axes to the new axes.

2.3

Combined Translation and Rotation

In general, simplifying the equation of a conic whose axes are tilted with respect to the coordinate axes and whose centre (or vertex) is away from the origin requires both operations together: first…

2.4

Removing Linear and Cross Terms from a Second-Degree Equation

A general second-degree equation in two variables has the form Two separate simplifications are usually needed to bring this to a standard form: removing the linear terms by translation, and removing…

+Exercise 2(a)i9 questions
  1. Q1When the origin is shifted to the point $(2,3)$ by translation of axes, find the transformed equation of the straight line $2x + 3y - 5 = 0$…Free
  2. Q2When the origin is shifted to $(-1,2)$ by translation of axes, find the transformed equation of $x^2+y^2+2x-4y+1=0$.Free
  3. Q3When the origin is shifted to $(3,-4)$ by translation of axes, find the transformed equation of $x^2+y^2-6x+8y+9=0$.Free
  4. Q4Find the angle through which the axes are to be rotated so as to remove the $xy$ term in the equation $x^2+4xy+y^2-2x+2y-6=0$.Preview
  5. Q5Find the angle through which the axes are to be rotated so as to remove the $xy$ term in the equation $x^2+2\sqrt{3}xy-y^2-8=0$.Preview
  6. Q6When the axes are rotated through an angle of $45^{\circ}$, find the new coordinates of the point $(4,-3)$.Preview
  7. Q7When the axes are rotated through an angle of $30^{\circ}$, find the new coordinates of the point $(0,2)$.Preview
  8. Q8Find the point to which the origin should be shifted so as to remove the first-degree (linear) terms from the equation $x^2+y^2-4x+6y-7=0$.Preview
  9. Q9Find the point to which the origin is to be shifted so as to remove the linear terms in the equation $2x^2+3y^2-4x+6y-1=0$.Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 7 questions7 questions
  1. Q1When the coordinate axes are rotated through an angle $\pi/6$, find the transformed equation of $x^2 + 2\sqrt{3}\,xy - y^2 = 2a^2$.Preview
  2. Q2When the axes are rotated through an angle $\alpha$, find the transformed equation of $x\cos\alpha + y\sin\alpha = p$.Preview
  3. Q3When the axes are rotated through an angle $45^{\circ}$, the transformed equation of a curve is $17x^2 - 16xy + 17y^2 = 225$. Find the origi…Preview
  4. Q4When the axes are rotated through an angle '$\alpha$', find the transformed equation of $x\cos\alpha + y\sin\alpha = P$.Preview
  5. Q5When the origin is shifted to the point $(2, 3)$ the transformed equation of a curve is $x^2 + 3xy - 2y^2 + 17x - 7y - 11 = 0$. Find the ori…Preview
  6. Q6When the axes are rotated through an angle $\dfrac{\pi}{6}$, find the transformed equation of $x^2 + 2\sqrt{3}\,xy - y^2 = 2a^2$.Preview
  7. Q7When the axes are rotated through an angle $\dfrac{\pi}{6}$, find the transformed equation of $x^2 + 2\sqrt{3}\,xy - y^2 = 2a^2$.Preview