CBSE Class 10 Coordinate geometry Sure Shot Questions Set F

Question. If the distance between the points (2, –2) and (–1, x) is 5, one of the values of x is
(a) –2
(b) 2
(c) –1
(d) 1
Answer: (b)
Explanation: According to question
\[ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = 5 \]
\[ \sqrt{(2+1)^2 + (-2-x)^2} = 5 \]
\[ \sqrt{9 + (-2-x)^2} = 5 \]
\[ 9 + (-2-x)^2 = 25 \]
\[ (2+x)^2 = 16 \]
\[ 2+x = 4 \]
\[ x = 2 \]

Question. The mid-point of the line segment joining the points A (–2, 8) and B (– 6, – 4) is
(a) (– 4, – 6)
(b) (2, 6)
(c) (– 4, 2)
(d) (4, 2)
Answer: (c)
Explanation: Let the coordinates of midpoint be (x, y) then
\[ x = \frac{-2-6}{2} \Rightarrow x = \frac{-8}{2} \Rightarrow x = -4 \]
\[ y = \frac{8-4}{2} \Rightarrow y = \frac{4}{2} \Rightarrow y = 2 \]
There fore the coordinates are (– 4, 2).

Question. The points A (9, 0), B (9, 6), C (–9, 6) and D (–9, 0) are the vertices of a
(a) Square
(b) Rectangle
(c) Rhombus
(d) Trapezium
Answer: (b)
Explanation: Here we will calculate the measure of all four sides of the quadrilateral fromed by given points A, B, C and D.
\[ AB = \sqrt{(9-9)^2 + (6-0)^2} = 6 \]
\[ BC = \sqrt{(9-(-9))^2 + (6-6)^2} = \sqrt{(18)^2} = 18 \]
\[ CD = \sqrt{(-9-(-9))^2 + (6-0)^2} = 6 \]
\[ AD = \sqrt{(9-(-9))^2 + (0-0)^2} = \sqrt{(18)^2} = 18 \]
Since, AB = CD and BC = AD. Therefore given points A,B,C and D are the vertices of a rectangle.

Question. The distance of the point P (2, 3) from the x-axis is
(a) 2
(b) 3
(c) 1
(d) 5
Answer: (b)
Explanation: Distance of the point P (2, 3) from the x-axis = Ordinate of the point (2, 3) i.e. 3.

Question. The distance between the points A (0, 6) and B (0, –2) is
(a) 6
(b) 8
(c) 4
(d) 2
Answer: (b)
Explanation: Here, \( x_1 = 0, y_1 = 6, x_2 = 0, y_2 = –2 \)
Distance between points AB = \( \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \)
AB = \( \sqrt{(0-0)^2 + (-2-6)^2} = 8 \)

Question. AOBC is a rectangle whose three vertices are vertices A (0, 3), O (0, 0) and B (5, 0). The length of its diagonal is
(a) 5
(b) 3
(c) \( \sqrt{34} \)
(d) 4
Answer: (c)
Explanation: The length of the diagonal is distance between the points AB. The distance is calculated as,
\[ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = \sqrt{(5-0)^2 + (0-3)^2} = \sqrt{34} \]

Question. If P (a/3, 4) is the mid-point of the line segment joining the points Q (– 6, 5) and R (– 2, 3), then the value of a is
(a) – 4
(b) – 12
(c) 12
(d) – 6
Answer: (b)
Explanation: As (a/3, 4) is the mid – point of the line segment joining the points Q (– 6, 5) and R (– 2, 3). Therefore
\[ \frac{-6-2}{2} = \frac{a}{3} \Rightarrow \frac{-8}{2} = \frac{a}{3} \Rightarrow -4 = \frac{a}{3} \Rightarrow a = -12 \]

Question. A circle drawn with origin as the centre passes through \( (\frac{13}{2}, 0) \). The point which does not lie in the interior of the circle is
(a) \( (-\frac{3}{4}, 1) \)
(b) \( (2, \frac{7}{3}) \)
(c) (5, –1/2)
(d) (–6, 5/2)
Answer: (d)
Explanation: If the point lies in the interior of circle, the distance of the point from the centre should be less than radius of circle. The radius of circle is the distance between origin and the point \( (\frac{13}{2}, 0) \) which is 6.5.
Distance between origin and (-3/4, 1) is \( \sqrt{(-\frac{3}{4}-0)^2 + (1-0)^2} = \sqrt{\frac{9}{16}+1} = \frac{5}{4} = 1.25 < 6.5 \).
Similarly the distance of points (2, 7/3) and (5, –1/2) is also less than 6.5.
But the distance of (–6, 5/2) is equal to 6.5. So the point (–6, 5/2) does not lie in the interior of circle.

Question. If the distance between the points (4, p) and (1, 0) is 5, then the value of p is
(a) 4 only
(b) ± 4
(c) – 4 only
(d) 0
Answer: (b)
Explanation: According to question:
\[ \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = 5 \Rightarrow \sqrt{(4-1)^2 + (p-0)^2} = 5 \]
\[ \Rightarrow \sqrt{9 + p^2} = 5 \Rightarrow 9 + p^2 = 25 \Rightarrow p^2 = 16 \Rightarrow p = \pm 4 \]

Question. The area of a triangle with vertices A (3, 0), B (7, 0) and C (8, 4) is:
(a) 14
(b) 28
(c) 8
(d) 6
Answer: (c)
Explanation: Area of triangle is calculated as,
\[ \text{Area} = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \]
\[ \Rightarrow \text{Area} = \frac{1}{2} |3(0-4) + 7(4-0) + 8(0-0)| \]
\[ \Rightarrow \text{Area} = \frac{1}{2} |-12 + 28| = \frac{1}{2} |16| = 8 \]

Question. The point which divides the line segment joining the points (7, –6) and (3, 4) in ratio 1 : 2 internally lies in the
(a) I quadrant
(b) II quadrant
(c) III quadrant
(d) IV quadrant
Answer: (d)
Explanation: Let the point be (x, y). Then, by using section formula:
\[ x = \frac{1(3)+2(7)}{1+2} = \frac{17}{3} \]
\[ y = \frac{1(4)+2(-6)}{1+2} = \frac{-8}{3} \]
Therefore, the point is (17/3, -8/3) which lies in fourth quadrant.

Question. One of the two points of trisection of the line segment joining the points A (7, – 2) and B (1, – 5) which divides the line in the ratio 1:2 are:
(a) (5, –3)
(b) (5, 3)
(c) (–5, –3)
(d) (13, 0)
Answer: (a)
Explanation: Required point of trisection that divides the given line in the ratio 1: 2 is
\[ (\frac{1(1)+2(7)}{1+2}, \frac{1(-5)+2(-2)}{1+2}) = (\frac{15}{3}, \frac{-9}{3}) = (5, -3) \]

Question. A line intersects the y-axis and x-axis at the points 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)
Answer: (d)
Explanation: As the line intersects the y and x axis, let the coordinates be (0, b) and (a, 0) respectively. Since (2, –5) is the midpoint of the axis. Therefore,
\[ 2 = \frac{a+0}{2} \Rightarrow a = 4 \]
\[ -5 = \frac{b+0}{2} \Rightarrow b = -10 \]
Therefore, the coordinates are (0, –10) and (4, 0).

Question. The ratio in which the point P (3/4, 5/12) divides the line segment joining the Points A (1/2, 3/2) and B (2, –5) is:
(a) 1:5
(b) 5:1
(c) 1:3
(d) 3:1
Answer: (a)
Explanation: Let the ratio be m : n then, according to the question:
\[ \frac{3}{4} = \frac{m(2)+n(\frac{1}{2})}{m+n} \Rightarrow \frac{3}{4}m + \frac{3}{4}n = 2m + \frac{1}{2}n \]
\[ \Rightarrow \frac{3}{4}n - \frac{1}{2}n = 2m - \frac{3}{4}m \Rightarrow \frac{1}{4}n = \frac{5}{4}m \Rightarrow n = 5m \Rightarrow m:n = 1:5 \]

Question. In which quadrant does the point (–10, 2) lie?
Answer: 2nd Quad.

Question. (0, 2) and (0, –5) are the co-ordinate of two points lying on _________ axis.
Answer: y

Question. Find the distance of a point P(x, y) from the origin (0, 0).
Answer: \( \sqrt{x^2 + y^2} \)

Question. What are the coordinates of mid-point of line joining the points (6, –2) and (4, 8)?
Answer: (5, 3)

Question. What are the coordinate of centroid of trinangle formed by the points (–7, 6), (8, 5), (2, –2)?
Answer: (1, 3)

Question. What is the x - coordinate of the point which divides the line joining (1, 2) and (2, 3) in the ratio 4 : 3?
Answer: \( \frac{11}{7} \)

Question. Find the coordinates of points which divides line joining (–4, 0) and (0, 6) in the ratio 1 : 3.
Answer: (–3, 1.5)

Question. Find the third vertex of a triangle if two of its vertices are (–1, 4) and (5, 2) and centroid is (0, –3).
Answer: (–4, –15)

Question. What is the length of the line AB, where A (1, 0) and B (5, 3).
Answer: 5

Question. Find the length of the line AB, where coordinates of points A and B are (2, 7) and (–2, 4).
Answer: 5

Question. What is the distance of the point (8, –2) from the origin?
Answer: \( \sqrt{68} \)

Question. Find the centroid of a triangle whose vertices are (–2, –3), (–1, 0) and (7, –6)
Answer: \( (\frac{4}{3}, -3) \)

Question. What is distance between points ( –3, 2) and (1, –2)?
Answer: \( 4\sqrt{2} \)

Question. What is the area of triangle ABC, whose vertices are \( A(x_1, y_1) \), \( B(x_2, y_2) \) and \( C(x_3, y_3) \)
Answer: \( \frac{1}{2} [x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)] \)

Question. Find the coordinates of points dividing the points (3, 5) and (7, 9) in the ratio 2 : 3.
Answer: \( (\frac{23}{5}, \frac{33}{5}) \)

Question. Find the distance of the point (0, 2) from the mid-point of the line joining (4, 10) and (2, 2).
Answer: 5

Question. Point Q lies on the line joining origin and P in such a way that OP = OQ. What will be the co-ordinates of Q, if coordinates of P are (–3, 2).
Answer: externally (3, –2), (–3, 2)

Question. If AB is the line joining (0, 1) and (4, –2) and CD is the line joining (1, 2) and (6, 4), then what is \( CD^2 – AB^2 \)?
Answer: 36

Question. Find the ratio in which the line segement joining (–2, –3) and (5, 6) is divided by x-axis.
Answer: 1 : 2

Question. Find the sum of lengths of the diagonals AC and BD of quadrilateral ABCD if A(3, 0), B(5, 3), C(0, 7) and D(–2, 0).
Answer: \( 2\sqrt{58} \)

Question. Find the sum of lengths of AB and BC if the coordinates of A, B and C are (1, 2), (–2, –2) and (4, 6) respectively.
Answer: 15

Question. Find the value of k if the point (0, 3) is equidistant from (5, k) and (k, k).
Answer: 5

Question. What is the distance between the points (–2, 4) and (–4, 3)?
Answer: \( \sqrt{5} \)

Question. What is the area of triangle formed by the points (3, 0), (0, 4), and (0, 0)?
Answer: 6sq. units

Question. Find the coordinates of a point, which is at a distance of 13 units from the origin and lies on x-axis.
Answer: (13, 0) or (–13, 0)

Question. One end of a diameter of a circle is (2, 3) and the centre is (–2, 5). What are the coordinates of the other end of his diameter?
Answer: (–6, 7)

Question. Find the length of median AD of the triangle formed by the points A(0, 6), B(8, 0) and C(4, 2).
Answer: \( \sqrt{61} \)

Question. Gunjan walks 12m due east and then 5m due north. At what distance is Gunjan from the starting point?
Answer: 13m

Question. The base BC of an equilateral triangle ABC with side 10cm lies along x=axis such that the mid-point of the base is at the origin. Find the coordinates of point B.
Answer: (–5, 0)

Question. Find the coordinates of fourth vertex of the rectangle formed by the points (0, 0), (2, 0) and (0, 3).
Answer: (2, 3)

Question. Find the value of x such that Q is the mid-point of PR and coordinates of P, Q and R are (6, –2), (1, 3) and (x, 8) respectively.
Answer: –4

Question. The line segment joining the points (3, –4) and (1, 2) is trisected at the points P and Q. Find the coordinates of P.
Answer: \( (\frac{7}{3}, –2) \)

Question. Find the value of y if the points A(5, y), B(1, 5), C(2, 1) and D(6, 2) are the vertices of the square.
Answer: 6

Question. A(3, 2) and B(–2, 1) are two vertices of a \( \Delta ABC \) whose centroid G has coordinates \( (\frac{5}{3}, \frac{-1}{3}) \). Find the coordinates of the third vertex C of the triangle.
Answer: (4, –4)

Question. What is the area of \( \Delta ABC \) if points A, B and C are collinear?
Answer: zero

Question. Find the ratio in which the line-segements joining the points (6, 4) and (1, –7) is divided internally by the axis of x.
Answer: \( \frac{4}{7} \)

Question. The three vetices of a rhombus taken in order are (–2, –1), (3, 4) and (–2, 3). Find the fourth vertex.
Answer: (–7, –2)

Question. Find the third vertex of a triangle, if two of its vertices are (–3, 1) and (0, –2) the centroid is at origin.
Answer: (3, 1)

Question. The mid-point of the line segement joining (3p, 4) and (–2, 2q) is (2, 6). Find the value of p.
Answer: 2

Question. In which ratio is the line joining the points A(–4, 4) and B(7, 7) divided by (0, –1)?
Answer: \( \frac{4}{7} \)

Question. P is the point of x-axis such that its distance from the origin is 3 units. Find the point Q on y-axis such that OP = OQ.
Answer: (0, 3) or (0, –3)

Question. The line segment joining the points (–4, 0) and (0, 6) is divided into four equal parts at P, Q and R. Find the coordinates of Q.
Answer: (–2, 3)

Question. Find the value of k if the distance between (k, 5) and (4, 5) is 5.
Answer: 9 or (–1)

Question. The points (0, –1), (2, 1), (0, 3) and (–2, 1) are the corners of a square. Find the length of its side.
Answer: \( \sqrt{8} \)

Question. If (p, q) is the mid-point of (5, 3) and (–2, –4) find the value of p + q.
Answer: 1

Question. Find the coordinates of the fourth point B such that OABC forms a square and coordinates of O, A and C being (0, 0), (3, 0) and (0, 3).
Answer: (3, 3)

Question. If two adjacent vertices of parallelogram are (3, 2), (–1, 0) and the diagonals cut at (2, –5) find the coordinates of other vertices of the parallelogram.
Answer: (1, –12), (5, –10)

Question. Find the coordinates of a point on x-axis which is equidistant from (–2, 5) and (2, –3).
Answer: (–2, 0)