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

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

Ex. 22.10Ex. 22.20Ex. 22.30Others

#### Chapter 22: Brief review of cartesian system of rectangular co-ordinates Exercise 22.10 solutions [Pages 12 - 13]

Ex. 22.10 | Q 1 | Page 12

If the line segment joining the points P (x1, y1) and Q (x2, y2) subtends an angle α at the origin O, prove that
OP · OQ cos α = x1 x2 + y1, y2

Ex. 22.10 | Q 2 | Page 13

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

Ex. 22.10 | Q 3 | Page 13

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.

Ex. 22.10 | Q 4 | Page 13

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.

Ex. 22.10 | Q 5 | Page 13

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

Ex. 22.10 | Q 6 | Page 13

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.

Ex. 22.10 | Q 7 | Page 13

Find the distance between P (x1, y1) and Q (x2, y2) when (i) PQ is parallel to the y-axis (ii) PQ is parallel to the x-axis.

Ex. 22.10 | Q 8 | Page 13

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

#### Chapter 22: Brief review of cartesian system of rectangular co-ordinates Exercise 22.20 solutions [Page 18]

Ex. 22.20 | Q 1 | Page 18

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

Ex. 22.20 | Q 2 | Page 18

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.

Ex. 22.20 | Q 3 | Page 18

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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Ex. 22.20 | Q 4 | Page 18

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

Ex. 22.20 | Q 5 | Page 18

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

Ex. 22.20 | Q 6 | Page 18

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

Ex. 22.20 | Q 7 | Page 18

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.

Ex. 22.20 | Q 8 | Page 18

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.

Ex. 22.20 | Q 9 | Page 18

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.

Ex. 22.20 | Q 10 | Page 18

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.

Ex. 22.20 | Q 11 | Page 18

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.

Ex. 22.20 | Q 12 | Page 18

If O is the origin and Q is a variable point on y2 = x, find the locus of the mid-point of OQ.

#### Chapter 22: Brief review of cartesian system of rectangular co-ordinates Exercise 22.30 solutions [Page 21]

Ex. 22.30 | Q 1 | Page 21

What does the equation (x − a)2 + (y − b)2 = r2 become when the axes are transferred to parallel axes through the point (a − c, b)?

Ex. 22.30 | Q 2 | Page 21

What does the equation (a − b) (x2 + y2) −2abx = 0 become if the origin is shifted to the point $\left( \frac{ab}{a - b}, 0 \right)$ without rotation?

Ex. 22.30 | Q 3.1 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
x2 + xy − 3x − y + 2 = 0

Ex. 22.30 | Q 3.2 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
x2 − y2 − 2x +2y = 0

Ex. 22.30 | Q 3.3 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − x − y + 1 = 0

Ex. 22.30 | Q 3.4 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y2 − x + y = 0

Ex. 22.30 | Q 4 | Page 21

To what point should the origin be shifted so that the equation x2 + xy − 3x − y + 2 = 0 does not contain any first degree term and constant term?

Ex. 22.30 | Q 5 | Page 21

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).

Ex. 22.30 | Q 6.1 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
x2 + xy − 3y2 − y + 2 = 0

Ex. 22.30 | Q 6.2 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − y2 − x + y = 0

Ex. 22.30 | Q 6.3 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
xy − x − y + 1 = 0

Ex. 22.30 | Q 6.4 | Page 21

Find what the following equation become when the origin is shifted to the point (1, 1).
x2 − y2 − 2x + 2y = 0

Ex. 22.30 | Q 7.1 | Page 21

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:  y2 + x2 − 4x − 8y + 3 = 0

Ex. 22.30 | Q 7.2 | Page 21

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: x2 + y2 − 5x + 2y − 5 = 0

Ex. 22.30 | Q 7.3 | Page 21

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: x2 − 12x + 4 = 0

Ex. 22.30 | Q 8 | Page 21

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).

#### Chapter 22: Brief review of cartesian system of rectangular co-ordinates solutions [Pages 21 - 22]

Q 1 | Page 21

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

Q 2 | Page 21

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

Q 3 | Page 21

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

Q 4 | Page 21

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

Q 5 | Page 22

If the points (a, 0), (at12, 2at1) and (at22, 2at2) are collinear, write the value of t1 t2.

Q 6 | Page 22

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.

Q 7 | Page 22

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

Q 8 | Page 22

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

Q 9 | Page 22

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.

Q 10 | Page 22

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

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

Ex. 22.10Ex. 22.20Ex. 22.30Others

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

RD Sharma solutions for Class 11 Maths chapter 22 (Brief review of cartesian system of rectangular co-ordinates) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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

Using RD Sharma Class 11 solutions Brief review of cartesian system of rectangular co-ordinates exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

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