## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 19 Coordinate Geometry

**EXERCISE 19.1**

**Question 1.**

**Find the co-ordinates of points whose**

**(i) abscissa is 3 and ordinate -4.**

**(ii) abscissa is – \(\frac { 3 }{ 2 }\)and ordinate 5.**

**(iii) whose abscissa is -1\(\frac { 2 }{ 3 }\) and ordinate -2 \(\frac { 1 }{ 4 }\) .**

**(iv) whose ordinate is 5 and abscissa is -2**

**(v) whose abscissa is -2 and lies on x-axis.**

**(vi) whose ordinate is \(\frac { 3 }{ 2 }\) and lies on y-axis.**

**Solution:**

**Question 2.**

**In which quadrant or on which axis each of the following points lie?**

**(-3, 5), (4, -1) (2, 0), (2, 2), (-3, -6)**

**Solution:**

**Question 3.**

**Which of the following points lie on**

**(i) x-axis? (ii) y-axis?**

**A (0, 2), B (5, 6), C (23, 0), D (0, 23), E (0, -4), F (-6, 0), G (√3,0)**

**Solution:**

**Question 4.**

**Plot the following points on the same graph paper :**

**A (3, 4), B (-3, 1), C (1, -2), D (-2, -3), E (0, 5), F (5, 0), G (0, -3), H (-3, 0).**

**Solution:**

**Question 5.**

Write the co-ordinates of the points A, B, C, D, E, F, G and H shown in the adjacent figure.

**Solution:**

**Question 6.**

**In which quadrants are the points A, B, C and D of problem 3 located ?**

**Solution:**

A Lies in the first quadrant, B lies on x-axis C lies in the third quadrant and D lies in the fourth quadrant.

**Question 7.**

**Plot the following points on the same graph paper :**

**Solution:**

**Question 8.**

**Plot the following points on the same graph paper.**

**Solution:**

**Question 9.**

**Plot the following points and check whether they are collinear or not:**

**(i) (1,3), (-1,-1) and (-2,-3)**

**(ii) (1,2), (2,-1) and (-1, 4)**

**(iii) (0,1), (2, -2) and (\(\frac { 2 }{ 3 }\) ,0)**

**Solution:**

**Question 10.**

**Plot the point P(-3, 4). Draw PM and PN perpendiculars to x-axis and y-axis respectively. State the co-ordinates of the points M and N.**

**Solution:**

**Question 11.**

**Plot the points A (1,2), B (-4,2), C (-4, -1) and D (1, -1). What kind of quadrilateral is ABCD ? Also find the area of the quadrilateral ABCD.**

**Solution:**

**Question 12.**

**Plot the points (0,2), (3,0), (0, -2) and (-3,0) on a graph paper. Join these points (in order). Name the figure so obtained and find the area of the figure obtained.**

**Solution:**

**Question 13.**

**Three vertices of a square are A (2,3), B(-3, 3) and C (-3, -2). Plot these points on a graph paper and hence use it to find the co-ordinates of the fourth vertex. Also find the area of the square.**

**Solution:**

**Question 14.**

**Write the co-ordinates of the vertices of a rectangle which is 6 units long and 4 units wide if the rectangle is in the first quadrant, its longer side lies on the x-axis and one vertex is at the origin.**

**Solution:**

**Question 15.**

**Repeat problem 12 assuming that the rectangle is in the third quadrant with all other conditions remaining the same.**

**Solution:**

A rectangle which is 6 unit long and 4 units wide and this rectangle is in the third quadrant.

**Question 16.**

**The adjoining figure shows an equilateral triangle OAB with each side = 2a units. Find the coordinates of the vertices.**

**Solution:**

**Question 17.**

**In the given figure, APQR is equilateral. If the coordinates of the points Q and R are (0, 2) and (0, -2) respectively, find the coordinates of the point P.**

**Solution:**

**EXERCISE 19.2**

**Question 1.**

**Draw the graphs of the following linear equations :**

**(i) 2x + + 3 = 0**

**(ii) x- 5y- 4 = 0**

**Solution:**

**Question 2.**

**Draw the graph of 3y= 12 – 2x. Take 2cm = 1 unit on both axes.**

**Solution:**

**Question 3.**

**Draw the graph of 5x + 6y – 30 = 0 and use it to find the area of the triangle formed by the line and the co-ordinate axes.**

**Solution:**

**Question 4.**

**Draw the graph of 4x- 3y + 12 = 0 and use it to find the area of the triangle formd by the line and the co-ordinate axes. Take 2 cm = 1 unit on both axes.**

**Solution:**

**Question 5.**

**Draw the graph of the equation y = 3x – 4. Find graphically.**

**(i) the value of y when x = -1**

**(ii) the value of x when y = 5.**

**Solution:**

**Question 6.**

**The graph of a linear equation in x and y passes through (4, 0) and (0, 3). Find the value of k if the graph passes through (A, 1.5).**

**Solution:**

**Question 7.**

**Use the table given alongside to draw the graph of a straight line. Find, graphically the values of a and b.**

**Solution:**

**EXERCISE 19.3**

**Question 1.**

**Solve the following equations graphically: 3x – 2y = 4, 5x – 2y = 0**

**Solution:**

**Question 2.**

**Solve the following pair of equations graphically. Plot at least 3 points for each straight line 2x – 7y = 6, 5x – 8y = – 4**

**Solution:**

**Question 3.**

**Using the same axes of co-ordinates and the same unit, solve graphically.**

**x+y = 0, 3x – 2y = 10**

**Solution:**

**Question 4.**

**Take 1 cm to represent 1 unit on each axis to draw the graphs of the equations 4x- 5y = -4 and 3x = 2y – 3 on the same graph sheet (same axes). Use your graph to find the solution of the above simultaneous equations.**

**Solution:**

**Question 5.**

**Solve the following simultaneous equations graphically, x + 3y = 8, 3x = 2 + 2y**

**Solution:**

**Question 6.**

**Solve graphically the simultaneous equations 3y = 5 – x, 2x = y + 3 (Take 2cm = 1 unit on both axes).**

**Solution:**

**Question 7.**

**Use graph paper for this question.**

**Take 2 cm = 1 unit on both axes.**

**(i) Draw the graphs of x +y + 3 = 0 and 3x-2y + 4 = 0. Plot only three points per line.**

**(ii) Write down the co-ordinates of the point of intersection of the lines.**

**(iii) Measure and record the distance of the point of intersection of the lines from the origin in cm.**

**Solution:**

**Question 8.**

**Solve the following simultaneous equations graphically :**

**2x-3y + 2 = 4x+ 1 = 3x – y + 2**

**Solution:**

**Question 9.**

**Use graph paper for this question.**

**(i) Draw the graphs of 3x -y – 2 = 0 and 2x + y – 8 = 0. Take 1 cm = 1 unit on both axes and plot three points per line.**

**(ii) Write down the co-ordinates of the point of intersection and the area of the traingle formed by the lines and the x-axis.**

**Solution:**

**Question 10.**

**Solve the following system of linear equations graphically : 2x -y – 4 = 0, x + y + 1 = 0. Hence, find the area of the triangle formed by these lines and the y-axis.**

**Solution:**

**Question 11.**

**Solve graphically the following equations: x + 2y = 4, 3x – 2y = 4**

**Take 2 cm = 1 unit on each axis. Write down the area of the triangle formed by the lines and the x-axis.**

**Solution:**

**Question 12.**

**On graph paper, take 2 cm to represent one unit on both the axes, draw the lines : x + 3 = 0, y – 2 = 0, 2x + 3y = 12 .**

**Write down the co-ordinates of the vertices of the triangle formed by these lines.**

**Solution:**

**Question 13.**

**Find graphically the co-ordinates of the vertices of the triangle formed by the lines y = 0, y – x and 2x + 3y= 10. Hence find the area of the triangle formed by these lines.**

**Solution:**

**EXERCISE 19.4**

**Question 1.**

**Find the distance between the following pairs of points :**

**(i) (2, 3), (4, 1)**

**(ii) (0, 0), (36, 15)**

**(iii) (a, b), (-a, -b)**

**Solution:**

**Question 2.**

**A is a point on y-axis whose ordinate is 4 and B is a point on x-axis whose abscissa is -3. Find the length of the line segment AB.**

**Solution:**

**Question 3.**

**Find the value of a, if the distance between the points A (-3, -14) and B (a, -5) is 9 units.**

**Solution:**

**Question 4.**

**(i) Find points on the x-axis which are at a distance of 5 units from the point (5, -4).**

**(ii) Find points on the y-axis are at a distance of 10 units from the point (8, 8) ?**

**(iii) Find points (or points) which are at a distance of √10 from the point (4, 3) given that the ordinate of the point or points is twice the abscissa.**

**Solution:**

**Question 5.**

**Find the point on the x-axis which, is equidistant from the points (2, -5) and (-2, 9).**

**Solution:**

**Question 6.**

**Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1, 3) and (x, 8) respectively.**

**Solution:**

**Question 7.**

**If Q (0, 1) is equidistant from P (5, -3) and R (x, 6) find the values of x.**

**Solution:**

**Question 8.**

**Find a relation between x and y such that the point (x, y) is equidistant from the points (7, 1) and (3, 5).**

**Solution:**

**Question 9.**

**The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from the points Q (2, -5) and U (-3, 6), then find the coordinates of P.**

**Solution:**

**Question 10.**

**If the points A (4,3) and B (x, 5) are on a circle with centre C (2, 3), find the value of x.**

**Solution:**

**Question 11.**

**If a point A (0, 2) is equidistant from the points B (3, p) and C (p, 5), then find the value of p.**

**Solution:**

**Question 12.**

**Using distance formula, show that (3, 3) is the centre of the circle passing through the points (6, 2), (0, 4) and (4, 6).**

**Solution:**

**Question 13.**

**The centre of a circle is C (2α – 1, 3α + 1) and it passes through the point A (-3, -1). If a diameter of the circle is of length 20 units, find the value(s) of α.**

**Solution:**

**Question 14.**

**Using distance formula, show that the points A (3, 1), B (6, 4) and C (8, 6) are coliinear.**

**Solution:**

**Question 15.**

**Check whether the points (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.**

**Solution:**

**Question 16.**

**Name the type of triangle formed by the points A (-5, 6), B (-4, -2) and (7, 5).**

**Solution:**

**Question 17.**

**Show that the points (1, 1), (- 1, – 1) and (-√3,√3) form an equilateral triangle.**

**Solution:**

**Question 18.**

**Show that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right triangle.**

**Solution:**

**Question 19.**

**The points A (0, 3), B (- 2, a) and C (- 1, 4) are the vertices of a right angled triangle at A, find the value of a.**

**Solution:**

**Question 20.**

**Show that the points (0, – 1), (- 2, 3), (6, 7) and (8, 3), taken in order, are the vertices of a rectangle. Also find its area.**

**Solution:**

**Question 21.**

**If P (2, -1), Q (3, 4), R (-2, 3) and S (-3, -2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus.**

**Solution:**

**Question 22.**

**Prove that the points A (2, 3), B {-2, 2), C (-1, -2) aqd D (3, -1) are the vertices of a square ABCD.**

**Solution:**

**Question 23.**

**Name the type of quadrilateral formedby the following points and give reasons for your answer :**

**(i) (-1, -2), (1, 0), (-1, 2), (-3, 0)**

**(ii) (4, 5), (7, 6), (4, 3), (1, 2)**

**Solution:**

**Question 24.**

**Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), (8, -2) and (2, -2). Also, find its circumradius.**

**Solution:**

**Question 25.**

**If two opposite vertices of a square are (3, 4) and (1, -1), find the coordinates of the other two vertices.**

**Solution:**

**Multiple Choice Questions**

**Choose the correct answer from the given four options (1 to 16):**

**Question 1.**

**Point (-3, 5) lies in the**

**(a) first quadrant**

**(b) second quadrant**

**(c) third quadrant**

**(d) fourth quadrant**

**Solution:**

Point (-3, 5) lies in second quadrant,** (b)**

**Question 2.**

**Point (0, -7) lies**

**(a) on the x-axis**

**(b) in the second quadrant**

**(c) on the y-axis**

**(d) the fourth quadrant**

**Solution:**

Point (0, -7) lies on y-axis (as x = 0)** (c)**

**Question 3.**

**Abscissa of a point is positive in**

**I and II quadrants**

**I and IV quadrants**

**I quadrant only**

**II quadrant only**

**Solution:**

Abscissa of a point is positive in first and fourth quadrants.** (b)**

**Question 4.**

**The point which lies ony-axis at a distance of 5 units in the negative direction of y- axis is**

**(a) (0, 5)**

**(b) (5, 0)**

**(c) (0, -5)**

**(d) (-5, 0)**

**Solution:**

(0, -5) is the required point.** (c)**

**Question 5.**

**If the perpendicular distance of a point P from the x-axis is 5 units and the foot of perpendicular lies on the negative direction of x-axis, then the point P has**

**(a) x-coordinate = -5**

**(b) y-coordinate = 5 only**

**(c) y-coordinate = -5 only**

**(d) y-coordinate = 5 or -5**

**Solution:**

Perpendicular distance for a point P on x- axis in negative direction.

It will has y = 5 and x = -5 **(d)**

**Question 6.**

**The points whose abscissa and ordinate have different signs will lie in**

**(a) I and II quadrants**

**(b) II and III quadrants**

**(c) I and III quadrants**

**(d) II and IV quadrants**

**Solution:**

Point which has abscissa and ordinate having different signs will lie in second and fourth quadrants.** (d)**

**Question 7.**

**The points (-5, 2) and (2, -5) lie in**

**(a) same quadrant**

**(b) II and III quadrants respectively**

**(c) II and IV quadrants respectively**

**(d) IV and II quadrants respectively**

**Solution:**

Points (-5, 2) and (2, -5) lie in second and fourth quadrants respectively. **(b)**

**Question 8.**

**If P (-1,1), Q (3, -4), R (1, -1), S (-2, -3) and T (-4, 4) are plotted on the graph paper, then point(s) in the fourth quadrant are**

**(a) P and T**

**(b) Q and R**

**(c) S only**

**(d) P and R**

**Solution:**

Points P (-1, 1), Q (3, -4), R (1, -1), S (-2, -3) and T (-4, 4) are plotted on graph The points in 4th quadrant are Q and R **(b)**

**Question 9.**

**On plotting the points O (0, 0), A (3, 0), B (3, 4), C (0, 4) and joining OA, AB, BC and CO which of the following figure is obtained?**

**(a) Square**

**(b) Rectangle**

**(c) Trapezium**

**(d) Rhombus**

**Solution:**

On plotting the points O (0, 0), A (3, 0), B (3, 4), C (0, 4)

OA, AB, BC and CO are joined

The figure so formed will a rectangle** (b)**

**Question 10.**

**Which of the following points lie on the graph of the equation :**

**3x-5y + 7 = 0?**

**(a) (1, -2)**

**(b) (2, 1)**

**(c) (-1, 2)**

**(d) (1, 2)**

**Solution:**

**Question 11.**

**The pair of equation x – a and y = b graphically represents lines which are**

**(a) parallel**

**(b) intersecting at (b, a)**

**(c) coincident**

**(d) intersecting at (a, b)**

**Solution:**

x = a, y = 6

Which are intersecting at (a, b)** (d)**

**Question 12.**

**The distance of the point P (2, 3) from the x>axis is**

**(a) 2 units**

**(b) 3 units**

**(c) 1 unit**

**(d) 5 units**

**Solution:**

The distance of the point P (2, 3) from x- axis is 3 units (as y = 3). **(b)**

**Question 13.**

**The distance of the point P (-4, 3) from the y-axis is**

**(a) 5 units**

**(b) -4 units**

**(c) 4 units**

**(d) 3 units**

**Solution:**

The distance of the point P (-4, 3) from y- axis will be 4 units.** (c)**

**Question 14.**

**The distance of the point P (-6, 8) from the origin is**

**(a) 8 units**

**(b) 2\(\sqrt { 7 }\) units**

**(c) 10 units**

**(d) 6 units**

**Solution:**

**Question 15.**

**The distance between the points A (0, 6) and B (0, -2) is**

**(a) 6 units**

**(b) 8 units**

**(c) 4 units**

**(d) 2 units**

**Solution:**

**Question 16.**

**The distance between the points (0, 5) and (-5, 0) is**

**(a) 5 units**

**(b) 5\(\sqrt { 2 }\)units**

**(c) 2 \(\sqrt { 7 }\) units**

**(d) 10 units**

**Solution:**

The distance between the points (0, 5) and (-5, 0) is

**Question 17.**

**AOBC is a rectangle whose three vertices are A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is**

**(a) 5 units**

**(b) 3 units**

**(c) \(\sqrt { 34 }\) units**

**(d) 4 units**

**Solution:**

**Question 18.**

**If the distance between the points (2, -2) and (-1, x) is S units, then one of the value of x is**

**(a) -2**

**(b) 2**

**(c) -1**

**(d) 1**

**Solution:**

**Question 19.**

**The distance between the points (4, p) and (1, 0) is 5 units, then the value of p is**

**(a) 4 only**

**(b) -4 only**

**(c) ±4**

**(d) 0**

**Solution:**

**Question 20.**

**The points (-4, 0), (4, 0) and (0, 3) are the vertices of a**

**(a) right triangle**

**(b) isosceles triangle**

**(c) equilateral triangle**

**(d) scalene triangle**

**Solution:**

**Question 21.**

**The area of a square whose vertices are A (0, -2), B (3, 1), C (0, 4) and D (-3, 1) is**

**(a) 18 sq. units**

**(b) 15 sq. units**

**(c) \(\sqrt { 18 }\) sq. units**

**(d) \(\sqrt { 15 }\) sq. units**

**Solution:**

**Question 22.**

**In the given figure, the area of the triangle ABC is**

**(a) 15 sq. units**

**(b) 10 sq. units**

**(c) 7.5 sq. units**

**(d) 2.5 sq. units**

**Solution:**

**Question 23.**

**The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is**

**(a) 5 units**

**(b) 12 units**

**(c) 11 units**

**(d) 7 + \(\sqrt { 5 }\) units**

**Solution:**

**Question 24.**

**If A is a point on the .y-axis whose ordinate is 5 and B is the point (-3, 1), then the length of AB is**

**(a) 8 units**

**(b) 5 units**

**(c) 3 units**

**(d) 25 units**

**Solution:**

**Question 25.**

**The point A (9, 0), B (9, 6), C (-9, 6) and D (-9, 0) are the vertices of a**

**(a) rectangle**

**(b) square**

**(c) rhombus**

**(d) trapezium**

**Solution:**

**Chapter Test**

**Question 1.**

**Three vertices of a rectangle are A (2, -1), B (2, 7) and C(4, 7). Plot these points on a graph and hence use it to find the co-ordinates of the fourth vertex D Also find the co-ordinates of**

**(i) the mid-point of BC**

**(ii) the mid point of CD**

**(iii) the point of intersection of the diagonals. What is the area of the rectangle ?**

**Solution:**

**Question 2.**

**Three vertices of a parallelogram are A (3, 5), B (3, -1) and C (-1, -3). Plot these points on a graph paper and hence use it to find the coordinates of the fourth vertex D. Also find the coordinates of the mid-point of the side CD. What is the area of the parallelogram?**

**Solution:**

**Question 3.**

**Draw the graphs of the following linear equations.**

**(i) y = 2x – 1**

**(ii) 2x + 3y = 6**

**(iii) 2x – 3y = 4.**

**Also find slope and y-intercept of these lines.**

**Solution:**

**Question 4.**

**Draw the graph of the equation 3x – 4y = 12. From the graph, find :**

**(i) the value of y when x = -4**

**(ii) the value of x when y = 3.**

**Solution:**

**Question 5.**

**Solve graphically, the simultaneous equations: 2x – 3y = 7; x + 6y = 11.**

**Solution:**

**Question 6.**

**Solve the following system of equations graphically: x – 2y – 4 = 0, 2x + .y – 3 = 0.**

**Solution:**

**Question 7.**

**Using a scale of l cm to 1 unit for both the axes, draw the graphs of the following equations : 6y = 5x:+ 10,y = 5;c-15. From the graph, find**

**(i) the coordinates of the point where the two lines intersect.**

**(ii) the area of the triangle between the lines and the x-axis.**

**Solution:**

**Question 8.**

**Find, graphically, the coordinates of the vertices of the triangle formed by the lines : 8y – 3x + 7 = 0, 2x-y + 4 = 0 and 5x + 4y = 29.**

**Solution:**

**Question 9.**

**Find graphically the coordinates of the vertices of the triangle formed by the lines y-2 = 0,2y + x = 0 and y + 1 = 3(x – 2). Hence, find the area of the triangle formed by these lines.**

**Solution:**

**Question 10.**

**A line segment is of length 10 units and one of its end is (-2,3). If the ordinate of the other end is 9, find the abscissa of the other end.**

**Solution:**

**Question 11.**

**A (-4, -1), B (-1, 2) and C (a, 5) are the vertices of an isosceles triangle. Find the value of a, given that AB is the unequal side.**

**Solution:**

**Question 12.**

**If A (-3, 2), B (a, p) and C (-1, 4) are the vertices of an isosceles triangle, prove that α + β = 1, given AB = BC.**

**Solution:**

**Question 13.**

**Prove that the points (3, 0), (6, 4) and (-1, 3) are the vertices of a right angled isosceles triangle.**

**Solution:**

**Question 14.**

**(i) Show that the points (2, 1), (0,3), (-2, 1) and (0, -1), taken in order, are the vertices of a square. Also find the area of the square.**

**(ii) Show that the points (-3, 2), (-5, -5), (2, -3) and (4, 4), taken in order, are the vertices of rhombus. Also find its area. Do the given points form a square?**

**Solution:**

**Question 15.**

**The ends of a diagonal of a square have co-ordinates (-2, p) and (p, 2). Find p if the area of the square is 40 sq. units.**

**Solution:**

**Question 16.**

**What type of quadrilateral do the points A (2, -2), B (7, 3), C (11, -1) and D (6, -6), taken in the order, form?**

**Solution:**

**Question 17.**

**Find the coordinates of the centre of the circle passing through the three given points A (5, 1), B (-3, -7) and C (7, -1).**

**Solution:**