CBSE Class 10 Mathematics Coordinate Geometry Worksheet Set 09

Question. The distance between the points \( (2, 3) \) and \( (4, 1) \) is
(a) \( \sqrt{2} \)
(b) \( 2\sqrt{2} \)
(c) \( \sqrt{3} \)
(d) \( 3\sqrt{3} \)
Answer: (b) \( 2\sqrt{2} \)

Question. The distance between the points \( (–5, 7) \) and \( (–1, 3) \) is
(a) \( 4\sqrt{3} \)
(b) \( 4\sqrt{5} \)
(c) \( 4\sqrt{2} \)
(d) \( 4\sqrt{7} \)
Answer: (c) \( 4\sqrt{2} \)

Question. The value of \( a \), so that the point \( (4, a) \) lies on the line \( 3x – 2y = 5 \) is
(a) 2
(b) 3
(c) \( \frac{7}{2} \)
(d) \( \frac{5}{2} \)
Answer: (c) \( \frac{7}{2} \)

Question. \( (5, –2), (6, 4) \) and \( (7, –2) \) are the vertices of a/an
(a) Scalene triangle
(b) Equilateral triangle
(c) Isosceles triangle
(d) None of the options
Answer: (c) Isosceles triangle

Question. The point on the \( y \)-axis which is equidistant from \( (2, –5) \) and \( (–2, 9) \) is
(a) \( (0, 3) \)
(b) \( (0, 2) \)
(c) \( (0, 5) \)
(d) None of the options
Answer: (b) \( (0, 2) \)

Exercise 4.1

Question. The distance of the point \( P (–3, –4) \) from the \( x \)-axis (in units) is
(a) 3
(b) –3
(c) 4
(d) 5
Answer: (c) 4

Question. The distance of the point \( P (3, –4) \) from the origin is
(a) 7 units
(b) 5 units
(c) 4 units
(d) 3 units
Answer: (b) 5 units

Question. The distance between the points \( (3, –2) \) and \( (–3, 2) \) is
(a) \( \sqrt{52} \) units
(b) \( 4\sqrt{10} \) units
(c) \( 2\sqrt{10} \) units
(d) 40 units
Answer: (a) \( \sqrt{52} \) units

Question. The distance of a point from the origin is
(a) \( x^2 + y^2 \)
(b) \( x^2 – y^2 \)
(c) \( \sqrt{x^2 + y^2} \)
(d) None of the options
Answer: (c) \( \sqrt{x^2 + y^2} \)

Question. The distance between the points \( (–\frac{8}{5}, 2) \) and \( (\frac{2}{5}, 2) \) is
(a) 0 units
(b) 1 unit
(c) 2 units
(d) 5 units
Answer: (c) 2 units

Question. The perpendicular distance of \( A(5, 12) \) from the \( y \)-axis is
(a) 4
(b) 5
(c) 7
(d) 8
Answer: (b) 5

Question. The distance between \( (\sqrt{2} + 1, 2) \) and \( (1, 2 – \sqrt{2}) \) is
(a) 2
(b) 3
(c) 11
(d) 17
Answer: (a) 2

Question. The value of \( k \) for which the point \( (0, 2) \) is equidistant from two points \( (3, k) \) and \( (k, 5) \) is
(a) 1
(b) 2
(c) 5
(d) 9
Answer: (a) 1

Question. If points \( (a, 0), (0, b) \) and \( (1, 1) \) are collinear, then the value of \( \frac{1}{a} + \frac{1}{b} \) is
(a) 1
(b) 2
(c) 5
(d) 9
Answer: (a) 1

Question. If the distance between the points \( (4, k) \) and \( (1, 0) \) is 5, then what can be the possible values of \( k \)?
(a) \( k = 4 \)
(b) \( k = –4 \)
(c) \( k = \pm 4 \)
(d) None of the options
Answer: (c) \( k = \pm 4 \)

Question. The value(s) of \( x \), if the distance between the points \( A(0, 0) \) and \( B(x, –4) \) is 5 units, is
(a) \( \pm 2 \)
(b) \( \pm 3 \)
(c) \( \pm 4 \)
(d) \( \pm 5 \)
Answer: (b) \( \pm 3 \)

Question. If the point \( A(0, 2) \) is equidistant from the points \( B(3, p) \) and \( C(p, 5) \), the value of \( p \) is
(a) 5 units
(b) \( \sqrt{8} \) units
(c) \( \sqrt{10} \) units
(d) None of the options
Answer: (c) \( \sqrt{10} \) units

Question. Points \( A(2, –1), B(3, 4), C(–2, 3), D(–3, –2) \) are the vertices of a:
(a) Rectangle
(b) Square
(c) Rhombus
(d) None of the options
Answer: (c) Rhombus

Question. The value(s) of \( y \), if the distance between the points \( (2, y) \) and \( (– 4, 3) \) is 10, is
(a) 11
(b) –5
(c) Both (a) and (b)
(d) None of the options
Answer: (c) Both (a) and (b)

Question. The points \( (a, a), (–a, –a) \) and \( (–\sqrt{3}a, \sqrt{3}a) \) are the vertices of a/an
(a) Equilateral triangle
(b) Isosceles triangle
(c) Scalene triangle
(d) None of the options
Answer: (a) Equilateral triangle

Question. If the distance of \( P(x, y) \) from \( A(5, 1) \) and \( B(–1, 5) \) are equal, then \( 3x \) equals
(a) \( y \)
(b) \( \frac{y}{2} \)
(c) \( 2y \)
(d) \( \frac{2}{y} \)
Answer: (c) \( 2y \)

Question. The point on \( x \)-axis which is equidistant from the points \( (2, –2) \) and \( (–4, 2) \) is
(a) \( (1, 0) \)
(b) \( (2, 0) \)
(c) \( (0, 2) \)
(d) \( (–1, 0) \)
Answer: (d) \( (–1, 0) \)

Question. Assertion (A): The point \( (0, 4) \) lies on \( y \)-axis.
Reason (R): The \( x \)-coordinate on the point on \( y \)-axis is zero.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion (A): The value of \( y \) is 6, for which the distance between the points \( P(2, –3) \) and \( Q(10, y) \) is 10.
Reason (R): Distance between two given points \( A (x_1, y_1) \) and \( B (x_2, y_2) \) is given by \( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (d) Assertion (A) is false but reason (R) is true.

Section Formula

• Internal Division: If a point \( P(x, y) \) divides the line segment \( XY \) in the ratio \( m : n \) internally, then the coordinate of point \( P \) is given by \( \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \), where coordinates of points \( X \) and \( Y \) are \( X(x_1, y_1) \) and \( Y(x_2, y_2) \). This is known as section formula.
• Mid-point formula: If \( P(x, y) \) is the mid-point of \( AB \), \( A(x_1, y_1) \), \( B(x_2, y_2) \), then coordinates of \( P(x, y) = P\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
• External Division: If a point \( P(x, y) \) divides the line segment \( AB \), \( A(x_1, y_1) \), \( B(x_2, y_2) \) externally in the ratio \( m : n \), then the coordinates of \( P(x, y) \) are \( P\left( \frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n} \right) \).
• Coordinate of centroid of a triangle ABC, \( A(x_1, y_1), B(x_2, y_2), C(x_3, y_3) \) is \( \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) \).
• To find the ratio in which the join of two points is divided according to some given condition, then we take ratio as \( k : 1 \) as \( \left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1} \right) \). If \( k \) is positive, then it is internal division.

Question. The coordinates of the points P and Q are respectively \( (4, –3) \) and \( (–1, 7) \). The \( x \)-coordinate (abscissa) of a point R on the line segment PQ such that \( \frac{PR}{PQ} = \frac{3}{5} \), is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1

Question. The ratio in which the line \( 2x + y – 4 = 0 \) divides the line segment joining the points \( A(2, –2) \) and \( B(3, 7) \) is
(a) 3 : 5
(b) 2 : 9
(c) 5 : 7
(d) 4 : 5
Answer: (b) 2 : 9

Question. Let P and Q be the points of trisection of the line segment joining the points \( A(2, – 2) \) and \( B(–7, 4) \) such that P is nearer to A. The coordinates of P and Q respectively are
(a) \( (3, 2), (1, 9) \)
(b) \( (–4, 3), (5, 0) \)
(c) \( (–3, 7), (2, 7) \)
(d) \( (–1, 0), (–4, 2) \)
Answer: (d) \( (–1, 0), (–4, 2) \)

Question. If \( (1, 2), (4, y), (x, 6) \) and \( (3, 5) \) are the vertices of a parallelogram taken in order, \( x \) and \( y \) respectively are
(a) \( x = 6, y = 3 \)
(b) \( x = 3, y = 6 \)
(c) \( x = –6, y = –3 \)
(d) \( x = –3, y = –6 \)
Answer: (a) \( x = 6, y = 3 \)

Exercise 4.2

Question. The point which lies on the perpendicular bisector of the line segment joining the points \( A(–2, –5) \) and \( B(2, 5) \) is
(a) \( (0, 0) \)
(b) \( (0, 2) \)
(c) \( (2, 0) \)
(d) \( (– 2, 0) \)
Answer: (a) \( (0, 0) \)

Question. The mid-point of the line segment joining the points \( (–5, 7) \) and \( (–1, 3) \) is
(a) \( (–3, 7) \)
(b) \( (–3, 5) \)
(c) \( (–1, 5) \)
(d) \( (5, –3) \)
Answer: (b) \( (–3, 5) \)

Question. If \( (3, –6) \) is the mid-point of the line segment joining \( (0, 0) \) and \( (x, y) \), then the point \( (x, y) \) is
(a) \( (–3, 6) \)
(b) \( (6, –6) \)
(c) \( (6, –12) \)
(d) \( (\frac{3}{2}, -3) \)
Answer: (c) \( (6, –12) \)

Question. If the point \( C(–1, 2) \) divides internally the line segment joining \( A(2, 5) \) and \( B \) in ratio \( 3 : 4 \), the coordinates of \( B \) are
(a) \( B(5, 2) \)
(b) \( B(–5, –2) \)
(c) \( B(–7, 2) \)
(d) \( B(7, –2) \)
Answer: (b) \( B(–5, –2) \)

Question. The point which divides the line segment joining the points \( A(0, 5) \) and \( B(5, 0) \) internally in the ratio \( 2 : 3 \) is
(a) \( (2, 3) \)
(b) \( (3, 4) \)
(c) \( (–2, 3) \)
(d) \( (3, –5) \)
Answer: (a) \( (2, 3) \)

Question. The coordinates of a point A, where AB is a diameter of the circle with centre \( (–2, 2) \) and B is the point with coordinates \( (3, 4) \) are
(a) \( A(7, 0) \)
(b) \( A(5, 0) \)
(c) \( A(–5, 0) \)
(d) \( A(–7, 0) \)
Answer: (d) \( A(–7, 0) \)

Question. The ratio in which the line segment joining the points \( (6, 4) \) and \( (1, –7) \) is divided by \( x \)-axis is
(a) 4 : 7
(b) 4 : 5
(c) 4 : 9
(d) 4 : 11
Answer: (a) 4 : 7

Question. The value of \( a \), for which point \( P (\frac{a}{3}, 2) \) is the mid-point of the line segment joining the points \( Q(–5, 4) \) and \( R(–1, 0) \) is
(a) –5
(b) –7
(c) –9
(d) –11
Answer: (c) –9

Question. The ratio in which the \( y \)-axis divides the line segment joining the points \( (5, –6) \) and \( (–1, –4) \) is
(a) 1 : 5
(b) 5 : 1
(c) 1 : 7
(d) 7 : 1
Answer: (b) 5 : 1

Question. If \( A(1, 2), B(4, 3) \) and \( C(6, 6) \) are three vertices of parallelogram ABCD, coordinates of D are
(a) \( (3, 5) \)
(b) \( (2, 7) \)
(c) \( (4, 9) \)
(d) \( (3, 8) \)
Answer: (a) \( (3, 5) \)

Question. The coordinates of the point P which divides the join of \( A(–2, 5) \) and \( B(3, –5) \) in the ratio \( 2 : 3 \) are
(a) \( (1, 0) \)
(b) \( (2, 0) \)
(c) \( (3, 0) \)
(d) \( (0, 1) \)
Answer: (d) \( (0, 1) \)

Question. The centre of a circle is \( (2a, a – 7) \). The value of \( a \) if the circle passes through the point \( (11, –9) \) and has diameter \( 10\sqrt{2} \) units is
(a) \( a = 3 \) or 6
(b) \( a = 5 \) or 3
(c) \( a = 7 \) or 4
(d) None of the options
Answer: (b) \( a = 5 \) or 3

Question. The value(s) of \( x \) for which the distance between the points \( P(x, 4) \) and \( Q(9, 10) \) is 10 units, is
(a) 15 or 2
(b) 10 or 9
(c) 17 or 1
(d) None of the options
Answer: (c) 17 or 1

Question. \( P(–2, 5) \) and \( Q(3, 2) \) are two points. The coordinates of the point R on PQ such that \( PR = 2QR \) are
(a) \( R(\frac{4}{3}, 3) \)
(b) \( R(\frac{2}{3}, 5) \)
(c) \( R(\frac{1}{3}, 7) \)
(d) None of the options
Answer: (a) \( R(\frac{4}{3}, 3) \)

Question. The coordinates of the point which divides the line segment joining the points \( (4, –3) \) and \( (8, 5) \) in the ratio \( 3 : 1 \) internally are
(a) \( P(4, 3) \)
(b) \( P(7, 3) \)
(c) \( P(3, 5) \)
(d) \( P(7, 3) \)
Answer: (b) \( P(7, 3) \)

Question. If the point \( P(m, 3) \) lies on the line segment joining the points \( A(-\frac{2}{5}, 6) \) and \( B(2, 8) \), the value of \( m \) is
(a) 3
(b) 2
(c) –3
(d) –4
Answer: (d) –4

Question. Assertion (A): The point \( (–1, 6) \) divides the line segment joining the points \( (–3, 10) \) and \( (6, –8) \) in the ratio \( 2 : 7 \) internally.
Reason (R): Given three points, i.e. A, B, C form an equilateral triangle, then \( AB = BC = AC \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Question. Assertion (A): Mid-point of a line segment divides line in the ratio \( 1 : 1 \).
Reason (R): The ratio in which the point \( (–3, k) \) divides the line segment joining the points \( (–5, 4) \) and \( (–2, 3) \) is \( 1 : 2 \).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: (c) Assertion (A) is true but reason (R) is false.