CBSE Class 10 Mathematics Coordinate Geometry MCQs Set 07

Question. Ratio in which the line \(3x + 4y = 7\) divides the line segment joining the points (1, 2) and (-2, 1) is
(a) 3 : 5
(b) 4 : 6
(c) 4 : 9
(d) None of the optioms
Answer: C
\(\frac{3(1) + 4(2) - 7}{-(3(-2) + 4(1) - 7)} = \frac{4}{-(-9)} = \frac{4}{9}\)

Question. If the points (a, 0), (0, b) and (1, 1) are collinear, then \(\frac{1}{a} + \frac{1}{b}\) equals
(a) 1
(b) 2
(c) 0
(d) -1
Answer: A
Let the given points are \(A(a, 0)\), \(B(0, b)\) and \(C(1, 1)\). Since, \(A, B, C\) are collinear.
Hence, \(ar(\Delta ABC) = 0\)
\(\frac{1}{2} [a(b - 1) + 0(1 - 0) + 1(0 - b)] = 0\)
\(ab - a - b = 0\)
\(a + b = ab\)
\(\frac{a + b}{ab} = 1\)
\(\frac{1}{a} + \frac{1}{b} = 1\)

Question. If the points \(A(4, 3)\) and \(B(x, 5)\) are on the circle with centre \(O(2, 3)\), then the value of \(x\) is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: C
Since, \(A\) and \(B\) lie on the circle having centre \(O\).
\(OA = OB\)
\(\sqrt{(4 - 2)^2 + (3 - 3)^2} = \sqrt{(x - 2)^2 + (5 - 3)^2}\)
\(2 = \sqrt{(x - 2)^2 + 4}\)
\(4 = (x - 2)^2 + 4\)
\((x - 2)^2 = 0\)
\(x = 2\)

Question. The ratio in which the point \((2, y)\) divides the join of \((-4, 3)\) and \((6, 3)\) and hence the value of \(y\) is
(a) 2 : 3, \(y = 3\)
(b) 3 : 2, \(y = 4\)
(c) 3 : 2, \(y = 3\)
(d) 3 : 2, \(y = 2\)
Answer: C
Let the required ratio be \(k : 1\)
Then, \(2 = \frac{6k - 4(1)}{k + 1}\)
or \(k = \frac{3}{2}\)
The required ratio is \(\frac{3}{2} : 1\) or \(3 : 2\)
Also, \(y = \frac{3(3) + 2(3)}{3 + 2} = 3\)

Question. The point on the X-axis which if equidistant from the points \(A(-2, 3)\) and \(B(5, 4)\) is
(a) (0, 2)
(b) (2, 0)
(c) (3, 0)
(d) (-2, 0)
Answer: B
Let \(P(x, 0)\) be a point on X-axis such that, \(AP = BP\)
\(AP^2 = BP^2\)
\((x + 2)^2 + (0 - 3)^2 = (x - 5)^2 + (0 + 4)^2\)
\(x^2 + 4x + 4 + 9 = x^2 - 10x + 25 + 16\)
\(14x = 28\)
\(x = 2\)
Hence, required point = (2, 0)

Question. C is the mid-point of \(PQ\), if \(P\) is \((4, x)\), \(C\) is \((y, -1)\) and \(Q\) is \((-2, 4)\), then \(x\) and \(y\) respectively are
(a) -6 and 1
(b) -6 and 2
(c) 6 and -1
(d) 6 and -2
Answer: A
Since, \(C(y, -1)\) is the mid-point of \(P(4, x)\) and \(Q(-2, 4)\).
We have, \(y = \frac{4 - 2}{2} = 1\)
and \(\frac{x + 4}{2} = -1 \Rightarrow x = -6\)

Question. If three points (0, 0), \((3, \sqrt{3})\) and \((3, \lambda)\) form an equilateral triangle, then \(\lambda\) equals
(a) 2
(b) -3
(c) -4
(d) None of the optioms
Answer: D
Let the given points are \(A(0, 0)\), \(B(3, \sqrt{3})\) and \(C(3, \lambda)\).
Since, \(\Delta ABC\) is an equilateral triangle, therefore \(AB = AC\)
\(\sqrt{(3 - 0)^2 + (\sqrt{3} - 0)^2} = \sqrt{(3 - 0)^2 + (\lambda - 0)^2}\)
\(9 + 3 = 9 + \lambda^2\)
\(\lambda^2 = 3\)
\(\lambda = \pm \sqrt{3}\)

Question. If the area of the triangle formed by the points \((x, 2x)\), \((-2, 6)\) and \((3, 1)\) is 5 sq units, then \(x\) equals
(a) 2/3
(b) 3/5
(c) 3
(d) 5
Answer: A
We have, area = 5 sq units
\(\frac{1}{2} [x(6 - 1) - 2(1 - 2x) + 3(2x - 6)] = \pm 5\)
\(5x - 2 + 4x + 6x - 18 = \pm 10\)
\(15x - 20 = \pm 10\)
\(15x = 30\) or \(10\)
\(x = \frac{30}{15}\) or \(\frac{10}{15}\)
\(x = 2\) or \(\frac{2}{3}\)

Question. The point which divides the line joining the points \(A(1, 2)\) and \(B(-1, 1)\) internally in the ratio 1 : 2 is
(a) \((\frac{-1}{3}, \frac{5}{3})\)
(b) \((\frac{1}{3}, \frac{5}{3})\)
(c) (-1, 5)
(d) (1, 5)
Answer: B

Question. If \(x - 2y + k = 0\) is a median of the triangle whose vertices are at points \(A(-1, 3)\), \(B(0, 4)\) and \(C(-5, 2)\), then the value of \(k\) is
(a) 2
(b) 4
(c) 6
(d) 8
Answer: D
Coordinate of the centroid \(G\) of \(\Delta ABC\) = \((\frac{-1 + 0 - 5}{3}, \frac{3 + 4 + 2}{3}) = (-2, 3)\)
Since, \(G\) lies on the median, \(x - 2y + k = 0\)
So, \(G\) satisfy the equation, \(-2 - 2(3) + k = 0 \Rightarrow k = 8\)

FILL IN THE BLANK

Question. The point which divide the line segment joining the points \( (5, 4) \) and \( (-6, -7) \) in the ratio \( 1:3 \) internally lies in the .......... quadrant.
Answer: first

Question. Point \( (-4, 6) \) divide the line segment joining the points \( A(-6, 10) \) and \( B(3, -8) \) in the ratio ..........
Answer: \( 2:7 \)

Question. If the coordinates of the points \( P, Q, R \) and \( S \) are such that \( PQ = QR = RS = SP \) and \( PQ \neq QS \), then quadrilateral \( DEFG \) is a ..........
Answer: rhombus

Question. \( (1, 2), (4, y), (x, 6) \) and \( (3, 5) \) are the vertices of a parallelogram taken in order, then the value of \( x \) and \( y \) are ..........
Answer: \( (6, 3) \)

Question. Points \( (1, 5), (2, 3) \) and \( (-2, -11) \) are ..........
Answer: Non-collinear

Question. All the points equidistant from two given points \( A \) and \( B \) lie on the .......... of the line segment \( AB \).
Answer: perpendicular bisector

Question. \( (5, -2), (6, 4) \) and \( (7, -2) \) are the vertices of an .......... triangle.
Answer: isosceles

Question. The distance of a point from the \( y \)-axis is called its ..........
Answer: abscissa

Question. If \( x - y = 2 \) then point \( (x, y) \) is equidistant from \( (7, 1) \) and (..........)
Answer: \( (3, 5) \)

Question. If the co-ordinates of the points \( A, B, C \) and \( D \) are such that \( AB = BC = CD = DA \) and \( AC = BD \), then quadrilateral \( ABCD \) is a ..........
Answer: square

Question. Distance between \( (2, 3) \) and \( (4, 1) \) is ..........
Answer: \( 2\sqrt{2} \)

Question. The distance of a point from the \( x \)-axis is called its ..........
Answer: ordinate

Question. The fourth vertex \( D \) of a parallelogram \( ABCD \) whose three vertices are \( A(-2, 5), B(6, 9) \) and \( C(8, 5) \) is ..........
Answer: \( (0, 1) \)

TRUE/FALSE

Question. The distance of a point from the \( y \)-axis is its ordinate.
Answer: False

Question. Area of the triangle formed by the points \( P(-1.5, 3), Q(6, -2) \) and \( R(-3, 4) \) is 0.
Answer: True

Question. The abscissa of point in the third quadrant is always negative.
Answer: True

Question. The ratio in which the point \( (3, 5) \) divides the join of \( (1, 3) \) and \( (4, 6) \) is \( 2:1 \).
Answer: True

Question. There exists only one point equidistant from two given points.
Answer: False

Question. The distance between \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is \( \sqrt{(x_2 + x_1)^2 + (y_2 + y_1)^2} \).
Answer: False

Question. The centroid of a triangle divides each median in the ratio \( 2:1 \).
Answer: True

Question. The coordinates of the point \( P(x, y) \) which divides the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is \( \left( \frac{m_1x_2 - m_2x_1}{m_1 + m_2}, \frac{m_1y_2 - m_2y_1}{m_1 + m_2} \right) \).
Answer: False

Question. The mid-point of the line segment joining the points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
Answer: True

Question. The area of the triangle formed by the points \( (x_1, y_1), (x_2, y_2) \) and \( (x_3, y_3) \) is the numerical value of the expression \( \frac{1}{2} [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] \).
Answer: True

MATCHING QUESTIONS

DIRECTION : Each question contains statements given in two Columns which have to be matched. Statements (A, B, C, D) in Column-I have to be matched with statements (p, q, r, s) in Column-II.

Question. Column-II gives distance between pair of points given in Column-I.
Column-I
(A) \( (-5, 7), (-1, 3) \)
(B) \( (5, 6), (1, 3) \)
(C) \( (\sqrt{3} + 1, 1), (0, \sqrt{3}) \)
(D) \( (0, 0), (-\sqrt{3}, \sqrt{3}) \)
Column-II
(p) \( \sqrt{17} \)
(q) \( \sqrt{8} \)
(r) \( \sqrt{6} \)
(s) \( 4\sqrt{2} \)
Answer: (A) — s, (B) — p, (C) — q, (D) — r

Question. Column-II gives the coordinates of the point \( P \) that divides the line segment joining the points given in Column-I.
Column-I
(A) \( A(-1, 3) \) and \( B(-5, 6) \) internally in the ratio \( 1 : 2 \)
(B) \( A(-2, 1) \) and \( B(1, 4) \) internally in the ratio \( 2 : 1 \)
(C) \( A(-1, 7) \) and \( B(4, -3) \) internally in the ratio \( 2 : 3 \)
(D) \( A(4, -3) \) and \( B(8, 5) \) internally in the ratio \( 3 : 1 \)
Column-II
(p) \( (7, 3) \)
(q) \( (0, 3) \)
(r) \( (1, 3) \)
(s) \( (1, 0) \)
Answer: (A) — s, (B) — q, (C) — r, (D) — p

Question. Column-II gives the area of triangles whose vertices are given in Column-I.
Column-I
(A) \( (2, 3), (-1, 0), (2, -4) \)
(B) \( (-5, -1), (3, -5), (5, 2) \)
(C) \( (1, -1), (-4, 6), (-3, -5) \)
(D) \( (0, 0), (8, 0), (0, 10) \)
Column-II
(p) 40
(q) 24
(r) 32
(s) 10.5
Answer: (A) — s, (B) — r, (C) — q, (D) — p

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(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.

Question. Assertion : The point \( (0, 4) \) lies on y-axis.
Reason : The x co-ordinate on the point on y-axis is zero.
Answer: A
Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A). The x co-ordinate of the point \( (0, 4) \) is zero. Point \( (0, 4) \) lies on y-axis.

Question. Assertion : The value of \( y \) is 6, for which the distance between the points \( P(2, -3) \) and \( Q(10, y) \) is 10.
Reason : Distance between two given points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by \( AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Answer: D
Assertion (A) is false but reason (R) is true. \( PQ = 10 \Rightarrow PQ^2 = 100 \Rightarrow (10 - 2)^2 + (y + 3)^2 = 100 \Rightarrow (y + 3)^2 = 100 - 64 = 36 \Rightarrow y + 3 = \pm 6 \Rightarrow y = -3 \pm 6 \Rightarrow y = 3, -9 \).

Question. Assertion : If \( A(2a, 4a) \) and \( B(2a, 6a) \) are two vertices of a equilateral triangle \( ABC \) then the vertex \( C \) is given by \( (2a + \sqrt{3}a, 5a) \).
Reason : In equilateral triangle all the coordinates of three vertices can be rational.
Answer: C
Assertion (A) is true but reason (R) is false. Let, \( A(x_1, y_1), B(x_2, y_2) \) and \( C(x_3, y_3) \) are all rational coordinates, \( \text{ar} (\triangle ABC) = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| = \frac{\sqrt{3}}{4} [(x_1 - x_2)^2 + (y_1 - y_2)^2] \). LHS = rational, RHS = irrational. Hence, vertices cannot be all rational.

Question. Assertion : The point \( (-1, 6) \) divides the line segment joining the points \( (-3, 10) \) and \( (6, -8) \) in the ratio \( 2 : 7 \) internally.
Reason : Three points \( A, B, \) and \( C \) are collinear if area of \( \triangle ABC = 0 \).
Answer: B
Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A). Using section formula, we have \( -1 = \frac{k \times 6 + 1 \times (-3)}{k + 1} \Rightarrow -k - 1 = 6k - 3 \Rightarrow 7k = 2 \Rightarrow k = \frac{2}{7} \). Ratio be \( 2 : 7 \) internally.