#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 22 : Brief review of cartesian system of rectangular co-ordinates

#### Page 13

If the line segment joining the points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) subtends an angle α at the origin O, prove that

OP · OQ cos α = x_{1} x_{2} + y_{1}, y_{2}

The vertices of a triangle ABC are A (0, 0), B (2, −1) and C (9, 2). Find cos B.

Four points A (6, 3), B (−3, 5), C (4, −2) and D (x, 3x) are given in such a way that \[\frac{\Delta DBC}{\Delta ABC} = \frac{1}{2}\]. Find x.

The points A (2, 0), B (9, 1), C (11, 6) and D (4, 4) are the vertices of a quadrilateral ABCD. Determine whether ABCD is a rhombus or not.

Find the coordinates of the centre of the circle inscribed in a triangle whose vertices are (−36, 7), (20, 7) and (0, −8).

The base of an equilateral triangle with side 2a lies along the y-axis, such that the mid-point of the base is at the origin. Find the vertices of the triangle.

Find the distance between P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) when (i) PQ is parallel to the y-axis (ii) PQ is parallel to the x-axis.

Find a point on the x-axis, which is equidistant from the points (7, 6) and (3, 4).

#### Page 18

Find the locus of a point equidistant from the point (2, 4) and the y-axis.

Find the equation of the locus of a point which moves such that the ratio of its distances from (2, 0) and (1, 3) is 5 : 4.

A point moves so that the difference of its distances from (ae, 0) and (−ae, 0) is 2a. Prove that the equation to its locus is \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]

Find the locus of a point such that the sum of its distances from (0, 2) and (0, −2) is 6.

Find the locus of a point which is equidistant from (1, 3) and the x-axis.

Find the locus of a point which moves such that its distance from the origin is three times its distance from the x-axis.

A (5, 3), B (3, −2) are two fixed points; find the equation to the locus of a point P which moves so that the area of the triangle PAB is 9 units.

Find the locus of a point such that the line segments with end points (2, 0) and (−2, 0) subtend a right angle at that point.

If A (−1, 1) and B (2, 3) are two fixed points, find the locus of a point P, so that the area of ∆PAB = 8 sq. units.

A rod of length *l* slides between two perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1 : 2.

Find the locus of the mid-point of the portion of the line x cos α + y sin α = p which is intercepted between the axes.

If O is the origin and Q is a variable point on y^{2} = x, find the locus of the mid-point of OQ.

#### Page 21

What does the equation (x − a)^{2} + (y − b)^{2} = r^{2} become when the axes are transferred to parallel axes through the point (a − c, b)?

What does the equation (a − b) (x^{2} + y^{2}) −2abx = 0 become if the origin is shifted to the point \[\left( \frac{ab}{a - b}, 0 \right)\] without rotation?

Find what the following equation become when the origin is shifted to the point (1, 1).

x^{2} + xy − 3x − y + 2 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).

x^{2} − y^{2} − 2x +2y = 0

Find what the following equation become when the origin is shifted to the point (1, 1).

xy − x − y + 1 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).

xy − y^{2} − x + y = 0

To what point should the origin be shifted so that the equation x^{2} + xy − 3x − y + 2 = 0 does not contain any first degree term and constant term?

Verify that the area of the triangle with vertices (2, 3), (5, 7) and (− 3 − 1) remains invariant under the translation of axes when the origin is shifted to the point (−1, 3).

Find what the following equation become when the origin is shifted to the point (1, 1).

x^{2} + xy − 3y^{2} − y + 2 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).

xy − y^{2} − x + y = 0

Find what the following equation become when the origin is shifted to the point (1, 1).

xy − x − y + 1 = 0

Find what the following equation become when the origin is shifted to the point (1, 1).

x^{2} − y^{2} − 2x + 2y = 0

Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: y^{2} + x^{2} − 4x − 8y + 3 = 0

Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: x^{2} + y^{2} − 5x + 2y − 5 = 0

Find the point to which the origin should be shifted after a translation of axes so that the following equation will have no first degree terms: x^{2} − 12x + 4 = 0

Verify that the area of the triangle with vertices (4, 6), (7, 10) and (1, −2) remains invariant under the translation of axes when the origin is shifted to the point (−2, 1).

#### Pages 21 - 22

The vertices of a triangle are O (0, 0), A (a, 0) and B (0, b). Write the coordinates of its circumcentre.

In Q.No. 1, write the distance between the circumcentre and orthocentre of ∆OAB.

Write the coordinates of the orthocentre of the triangle formed by points (8, 0), (4, 6) and (0, 0).

Three vertices of a parallelogram, taken in order, are (−1, −6), (2, −5) and (7, 2). Write the coordinates of its fourth vertex.

If the points (a, 0), (at_{1}^{2}, 2at_{1}) and (at_{2}^{2}, 2at_{2}) are collinear, write the value of t_{1} t_{2}.

If the coordinates of the sides AB and AC of ∆ABC are (3, 5) and (−3, −3), respectively, then write the length of side BC.

Write the coordinates of the circumcentre of a triangle whose centroid and orthocentre are at (3, 3) and (−3, 5), respectively.

Write the coordinates of the in-centre of the triangle with vertices at (0, 0), (5, 0) and (0, 12).

If the points (1, −1), (2, −1) and (4, −3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.

Write the area of the triangle with vertices at (a, b + c), (b, c + a) and (c, a + b).

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## RD Sharma solutions for Class 11 Mathematics chapter 22 - Brief review of cartesian system of rectangular co-ordinates

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Concepts covered in Class 11 Mathematics chapter 22 Brief review of cartesian system of rectangular co-ordinates are Brief Review of Cartesian System of Rectanglar Co-ordinates, Logarithmic Functions, Exponential Function, Pictorial Representation of a Function, Graph of Function, Pictorial Diagrams, Equality of Ordered Pairs, Ordered Pairs, Algebra of Real Functions, Some Functions and Their Graphs, Functions, Relation, Cartesian Product of Sets.

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