## RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry MCQS

These Solutions are part of RD Sharma Class 10 Solutions. Here we have given RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry MCQS

**Other Exercises**

- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.1
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.2
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.3
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.4
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry Ex 6.5
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry VSAQS
- RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry MCQS

**Mark the correct alternative in each of the following :**

**Question 1.**

The distance between the points (cosθ, sinθ) and (sinθ, -cosθ) is

(a) √3

(b) √2

(c) 2

(d) 1

**Solution:**

**(b)** Distance between (cosθ, sinθ) and (sinθ, -cosθ)

**Question 2.**

The distance between the points (a cos 25°, 0) and (0, a cos 65°) is

(a) a

(b) 2a

(c) 3a

(d) None of these

**Solution:**

**(a)** Distance between (a cos 25°, 0) and (0, a cos 65°)

**Question 3.**

If x is a positive integer such that the distance between points P (x, 2) and Q (3, -6) is 10 units, then x =

(a) 3

(b) -3

(c) 9

(d) -9

**Solution:**

**(c)** Distance between P (x, 2) and Q (3, -6) = 10 units

=> x (x – 9) + 3 (x – 9) = 0

(x – 9) (x + 3) = 0

Either x – 9 = 0, then x = 9 or x + 3 = 0, then x = -3

x is positive integer

Hence x = 9

**Question 4.**

The distance between the points (a cosθ + b sinθ, 0) and (0, a sinθ – b cosθ) is

(a) a² + b²

(b) a + b

(c) a² – b²

(d) √(a²+b²)

**Solution:**

**(d)** Distance between (a cosθ + b sinθ, 0) and (0, a sinθ – b cosθ)

**Question 5.**

If the distance between the points (4, p) and (1, 0) is 5, then p =

(a) ±4

(b) 4

(c) -4

(d) 0

**Solution:**

**(a)** Distance between (4, p) and (1, 0) = 5

**Question 6.**

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

(a) 9, 6

(b) 3, -9

(c) -3, 9

(d) 9, -6

**Solution:**

**(b)** Let the ordinate of other end = y

then distance between (2, -3) and (10, y) = 10 units

**Question 7.**

The perimeter of the triangle formed by the points (0, 0), (1, 0) and (0, 1) is

(a) 1 ± √2

(b) √2 + 1

(c) 3

(d) 2 + √2

**Solution:**

**(d)** Let the vertices of ∆ABC be A (0, 0), B(1, 0) and C (0, 1)

**Question 8.**

If A (2, 2), B (-4, -4) and C (5, -8) are the vertices of a triangle, then the length of the median through vertices C is

(a) √65

(b) √117

(c) √85

(d) √113

**Solution:**

**(c)** Let mid point of A (2, 2), B (-4, -4) be D whose coordinates will be

**Question 9.**

If three points (0, 0), (3, √3) and (3, λ) form an equilateral triangle, then λ =

(a) 2

(b) -3

(c) -4

(d) None of these

**Solution:**

**(d)** Let the points (0, 0), (3, √3) and (3, λ) from an equilateral triangle

AB = BC = CA

=> AB² = BC² = CA²

Now, AB² = (x_{2} – x_{1})² + (y_{2} – y_{1})²

**Question 10.**

If the points (k, 2k), (3k, 3k) and (3, 1) are collinear, then k

(a) \(\frac { 1 }{ 3 }\)

(b) \(\frac { -1 }{ 3 }\)

(c) \(\frac { 2 }{ 3 }\)

(d) \(\frac { -2 }{ 3 }\)

**Solution:**

**(b)** Let the points A (k, 2k), B (3k, 3k) and C (3, 1) be the vertices of a ∆ABC

**Question 11.**

The coordinates of the point on x-axis which are equidistant from the points (-3, 4) and (2, 5) are

(a) (20, 0)

(b) (-23, 0)

(c) (\(\frac { 4 }{ 5 }\) , 0)

(d) None of these

**Solution:**

**(d)**

**Question 12.**

If (-1, 2), (2, -1) and (3, 1) are any three vertices of a parallelogram, then

(a) a = 2, b = 0

(b) a = -2, b = 0

(c) a = -2, b = 6

(d) a = 0, b = 4

**Solution:**

**(d)** In ||gm ABCD, diagonals AC and AD bisect each other at O

O is mid-point of AC

**Question 13.**

If A (5, 3), B (11, -5) and P (12, y) are the vertices of a right triangle right angled at P, then y =

(a) -2, 4

(b) -2, 4

(c) 2, -4

(d) 2, 4

**Solution:**

**(c)** A (5, 3), B (11, -5) and P (12, y) are the vertices of a right triangle, right angle at P

**Question 14.**

The area of the triangle formed by (a, b + c), (b, c + a) and (c, a + b) is

(a) a + b + c

(b) abc

(c) (a + b + c)²

(d) 0

**Solution:**

**(d)** Vertices of a triangle are (a, b + c), (b, c + a) and (c, a + b)

**Question 15.**

If (x, 2), (-3, -4) and (7, -5) are coliinear, then x =

(a) 60

(b) 63

(c) -63

(d) -60

**Solution:**

**(c)** Area of triangle whose vertices are (x, 2), (-3, -4) and (7, -5)

**Question 16.**

If points (t, 2t), (-2, 6) and (3, 1) are collinear, then t =

(a) \(\frac { 3 }{ 4 }\)

(b) \(\frac { 4 }{ 3 }\)

(c) \(\frac { 5 }{ 3 }\)

(d) \(\frac { 3 }{ 5 }\)

**Solution:**

**(b)** The area of triangle whose vertices are (t, 2t), (-2, 6) and (3, 1)

**Question 17.**

If the area of the triangle formed by the points (x, 2x), (-2, 6) and (3, 1) is 5 square units, then x =

(a) \(\frac { 2 }{ 3 }\)

(b) \(\frac { 3 }{ 5 }\)

(c) 2

(d) 5

**Solution:**

**(c)** Area of triangle whose vertices are (x, 2x), (-2, 6) and (3, 1)

**Question 18.**

If points (a, 0), (0, b) and (1, 1) are collinear, then \(\frac { 1 }{ a }\) + \(\frac { 1 }{ b }\) =

(a) 1

(b) 2

(c) 0

(d) -1

**Solution:**

**(a)** The area of triangle whose vertices are (a, 0), (0, b) and (1, 1)

**Question 19.**

If the centroid of a triangle is (1, 4) and two of its vertices are (4, -3) and (-9, 7), then the area of the triangle is

(a) 183 sq. units

(b) \(\frac { 183 }{ 2 }\) sq. units

(c) 366 sq. units

(d) \(\frac { 183 }{ 4 }\) sq. units

**Solution:**

**(b)** Centroid of a triangle = (1, 4)

and two vertices of the triangle are (4, -3) and (-9, 7)

Let the third vertex be (x, y), then

= \(\frac { 183 }{ 2 }\) sq. units

**Question 20.**

The line segment joining points (-3, -4) and (1, -2) is divided by y-axis in the ratio

(a) 1 : 3

(b) 2 : 3

(c) 3 : 1

(d) 2 : 3

**Solution:**

**(c)** The point lies on y-axis

Its abscissa will be zero

Let the point divides the line segment joining the points (-3, -4) and (1, -2) in the ratio m : n

**Question 21.**

The ratio in which (4, 5) divides the join of (2, 3) and (7, 8) is

(a) -2 : 3

(b) -3 : 2

(c) 3 : 2

(d) 2 : 3

**Solution:**

**(d)** Let the point (4, 5) divides the line segment joining the points (2, 3) and (7, 8) in the ratio m : n

**Question 22.**

The ratio in which the X-axis divides the segment joining (3, 6) and (12, -3) is

(a) 2 : 1

(b) 1 : 2

(c) -2 : 1

(d) 1 : -2

**Solution:**

**(a)** The point lies on x-axis

Its ordinate is zero

Let this point divides the line segment joining the points (3, 6) and (12, -3) in the ratio m : n

**Question 23.**

If the centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is at the origin, then a^{3} + b^{3} + c^{3} =

(a) abc

(b) 0

(c) a + b + c

(d) 3 abc

**Solution:**

**(d)** Centroid of the triangle formed by the points (a, b), (b, c) and (c, a) is origin (0, 0)

**Question 24.**

If points (1, 2), (-5, 6) and (a, -2) are collinear, then a =

(a) -3

(b) 7

(c) 2

(d) -2

**Solution:**

**(b)** The area of a triangle whose vertices are (1, 2), (-5, 6) and (a, -2)

**Question 25.**

If the centroid of the triangle formed by (7, x), (y, -6) and (9, 10) is at (6, 3), then (x, y) =

(a) (4, 5)

(b) (5, 4)

(c) (-5, -2)

(d) (5, 2)

**Solution:**

**(d)** Centroid of (7, x), (y, -6) and (9, 10) is (6, 3)

**Question 26.**

The distance of the point (4, 7) from the x-axis is

(a) 4

(b) 7

(c) 11

(d) √65

**Solution:**

**(b)** The distance of the point (4, 7) from x-axis = 7

**Question 27.**

The distance of the point (4, 7) from the y-axis is

(a) 4

(b) 7

(c) 11

(d) √65

**Solution:**

**(a)** The distance of the point (4, 7) from y-axis = 4

**Question 28.**

If P is a point on x-axis such that its distance from the origin is 3 units, then the coordinates of a point Q on OY such that OP = OQ, are

(a) (0, 3)

(b) (3, 0)

(c) (0, 0)

(d) (0, -3)

**Solution:**

**(a)** P is a point on x-axis and its distance from 0 is 3

Co-ordinates of P will be (3, 0)

Q is a point on OY such that OP = OQ

Co-ordinates of Q will be (0, 3)

**Question 29.**

If the point (x, 4) lies on a circle whose centre is at the origin and radius is 5, then x =

(a) ±5

(b) ±3

(c) 0

(d) ±4

**Solution:**

**(b)** Point A (x, 4) is on a circle with centre O (0, 0) and radius = 5

**Question 30.**

If the point P (x, y) is equidistant from A (5, 1) and B (-1, 5), then

(a) 5x = y

(b) x = 5y

(c) 3x = 2y

(d) 2x = 3y

**Solution:**

**(c)**

**Question 31.**

If points A (5, p), B (1, 5), C (2, 1) and D (6, 2) form a square ABCD, then p =

(a) 7

(b) 3

(c) 6

(d) 8

**Solution:**

**(c)** Vertices of a square are A (5, p), B (1, 5), C (2, 1) and D (6, 2)

The diagonals bisect each other at O

O is the mid-point of AC and BD

O is mid-point of BD, then

**Question 32.**

The coordinates of the circumcentre of the triangle formed by the points O (0, 0), A (a, 0) and B (0, b) are

(a) (a, b)

(b) (\(\frac { a }{ 2 }\) , \(\frac { b }{ 2 }\))

(c) (\(\frac { b }{ 2 }\) , \(\frac { a }{ 2 }\))

(d) (b, a)

**Solution:**

**(b)** Let co-ordinates of C be (x, y) which is the centre of the circumcircle of ∆OAB

Radii of a circle are equal

**Question 33.**

The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (-3, 4) are

(a) (0, 2)

(b) (3, 0)

(b) (0, 3)

(d) (2, 0)

**Solution:**

**(d)** The given point P lies on x-axis

Let the co-ordinates of P be (x, 0)

The point P lies on the perpendicular bisector of of the line segment joining the points A (7, 6), B (-3, 4)

**Question 34.**

If the centroid of the triangle formed by the points (3, -5), (-7, 4), (10, -k) is at the point (k, -1), then k =

(a) 3

(b) 1

(c) 2

(d) 4

**Solution:**

**(c)** O (k, -1) is the centroid of triangle whose vertices are

**Question 35.**

If (-2, 1) is the centroid of the triangle having its vertices at (x, 0), (5, -2), (-8, y), then x, y satisfy the relation

(a) 3x + 8y = 0

(b) 3x – 8y = 0

(c) 8x + 3y = 0

(d) 8x = 3y

**Solution:**

(-2, 1) is the centroid of triangle whose vertices are (x, 0), (5, -2), (-8, y)

**Question 36.**

The coordinates of the fourth vertex of the rectangle formed by the points (0, 0), (2, 0), (0, 3) are

(a) (3, 0)

(b) (0, 2)

(c) (-2, 3)

(d) (3, 2)

**Solution:**

**(c)** Three vertices of a rectangle are A (0, 0), B (2, 0), C (0, 3)

Let fourth vertex be D (x, y)

The diagonals of a rectangle bisect eachother at O

O is the mid-point of AC, then

Coordinates of O will be (\(\frac { 0+0 }{ 2 }\) , \(\frac { 0+3 }{ 2 }\))

**Question 37.**

The length of a line segment joining A (2, -3) and B is 10 units. If the abscissa of B is 10 units, then its ordinates can be

(a) 3 or -9

(b) -3 or 9

(c) 6 or 27

(d) -6 or-27

**Solution:**

**(a)** Abscissa of B is 10 and co-ordinates of A are (2, -3)

Let ordinates of B be y, then

**Question 38.**

The ratio in which the line segment joining P(x_{1}, y_{1}) and Q (x_{2}, y_{2}) is divided by x-axis is

(a) y_{1} : y_{2}

(b) -y_{1} : y_{2}

(c) x_{1} : x_{2}

(d) -x_{1} : x_{2}

**Solution:**

**(b)** Let a point A on x-axis divides the line segment joining the points P (x_{1}, y_{1}), Q (x_{2}, y_{2}) in the ratio m_{1} : m_{2} and

let co-ordinates of A be (x, 0)

**Question 39.**

The ratio in which the line segment joining points A (a_{1}, b_{1}) and B (a_{2}, b_{2}) is divided by y-axis is

(a) -a_{1} : a_{2}

(b) a_{1} : a_{2}

(c) b_{1} : b_{2}

(d) -b_{1} : b_{2}

**Solution:**

**(a)** Let the point P on y-axis, divides the line segment joining the point A (a_{1}, b_{1}) and B (a_{2}, b_{2}) is the ratio m_{1} : m_{2} and

let the co-ordinates of P be (0, y), then

**Question 40.**

If the line segment joining the points (3, -4) and (1, 2) is trisected at points P

**Solution:**

**(b)**

**Question 41.**

If the coordinates of one end of a diameter of a circle are (2, 3) and the coordinates of its centre are (-2, 5), then the coordinates of the other end of the diameter are **[CBSE 2012]**

(a) (-6, 7)

(b) (6, -7)

(c) (6, 7)

(d) (-6, -7)

**Solution:**

**(a)** Let AB be the diameter of a circle with centre O

**Question 42.**

The coordinates of the point P dividing the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1 are

(a) (2, 4)

(b) (3, 5)

(c) (4, 2)

(d) (5, 3)** [CBSE 2012]**

**Solution:**

**(b)** Point P divides the line segment joining the points A (1, 3) and B (4, 6) in the ratio 2 : 1

Let coordinates of P be (x, y), then

**Question 43.**

In the figure, the area of ∆ABC (in square units) is **[CBSE 2013]**

(a) 15

(b) 10

(c) 7.5

(d) 2.5

**Solution:**

**(c)**

**Question 44.**

The point on the x-axis which is equidistant from points (-1, 0) and (5, 0) is

(a) (0, 2)

(b) (2, 0)

(c) (3, 0)

(d) (0, 3) **[CBSE 2013]**

**Solution:**

**(c)** Let the point P (x, 0) is equidistant from the points A (-1, 0), B (5, 0)

**Question 45.**

If A (4, 9), B (2, 3) and C (6, 5) are the vertices of ∆ABC, then the length of median through C is

(a) 5 untis

(b) √10 units

(c) 25 units

(d) 10 units **[CBSE 2014]**

**Solution:**

**(b)** A (4, 9), B (2, 3) and C (6, 5) are the vertices of ∆ABC

Let median CD has been drawn C (6, 5)

**Question 46.**

If P (2, 4), Q (0, 3), R (3, 6) and S (5, y) are the vertices of a paralelogram PQRS, then the value of y is

(a) 7

(b) 5

(c) -7

(d) -8 **[CBSE 2014]**

**Solution:**

**(a)** P (2, 4), Q (0, 3), R (3, 6) and S (5, y) are the vertices of a ||gm PQRS

**Question 47.**

If A (x, 2), B (-3, -4) and C (7, -5) are collinear, then the value of x is

(a) -63

(b) 63

(c) 60

(d) -60 **[CBSE 2014]**

**Solution:**

**(a)** A (x, 2), B (-3, -4) and C (7, -5) are collinear, then area ∆ABC = 0

Now area of ∆ABC

**Question 48.**

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

(a) 7 + √5

(b) 5

(c) 10

(d) 12 **[CBSE 2014]**

**Solution:**

**(d)** A (0, 4) and B (0, 0) and C (3, 0) are the vertices of ∆ABC

**Question 49.**

If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4), then

(a) AP = \(\frac { 1 }{ 3 }\) AB

(b) AP = BP

(c) BP = \(\frac { 1 }{ 3 }\) AB

(d) AP = \(\frac { 1 }{ 2 }\) AB

**Solution:**

**(d)** Given that, the point P (2, 1) lies on the line segment joining the points (4, 2) and B (8, 4), which shows in the figure below:

**Question 50.**

A line intersects the y-axis and x-axis at P and Q, respectively. If (2, -5) is the mid-point of PQ, then the coordinates of P and Q are, respectively

(a) (0, -5) and (2, 0)

(b) (0, 10) and (-4, 0)

(c) (0, 4) and (-10, 0)

(d) (0, -10) and (4, 0)

**Solution:**

**(d)** Let the coordinates of P and Q (0, y) and (x, 0), respectively.

So, the mid-point of P (0, y) and Q (x, 0) is

2 = \(\frac { x + 0 }{ 2 }\) and -5 = \(\frac { y + 0 }{ 2 }\)

=> 4 = x and -10 = y

=> x = 4 and y = -10

So, the coordinates of P and Q are (0, -10) and (4, 0).

Hope given RD Sharma Class 10 Solutions Chapter 6 Co-ordinate Geometry MCQS are helpful to complete your math homework.

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