Coordinate Geometry
Section A: Multiple Choice Questions
1. Coordinates of origin are:
Solution The origin is the intersection of the X and Y axes, where both coordinates are zero. Thus, (0, 0).
2. Which point lies on the y-axis?
Solution A point on the y-axis has its x-coordinate as 0. Among options, (0, 3) fits this condition.
3. Which point lies in the fourth quadrant?
Solution In the fourth quadrant, x is positive and y is negative (+, -). (3, -4) matches this.
4. Mid point of line joining (2, 4) and (6, 8) is:
Solution Midpoint = \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) = (\frac{2+6}{2}, \frac{4+8}{2}) = (\frac{8}{2}, \frac{12}{2}) = (4, 6)\).
5. The distance between the points P(0, 2) and Q(6, 0) is
Solution \(d = \sqrt{(6-0)^2 + (0-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}\).
6. What is y-coordinate of the mid-point (3, 7) and (5, 9)?
Solution \(y = \frac{7+9}{2} = \frac{16}{2} = 8\).
7. Find the length of the diagonal of a rectangle with vertices A(1, 2) B(1, 5) C(6, 5) D(6, 2):
Solution Diagonal AC length = \(\sqrt{(6-1)^2 + (5-2)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}\).
8. Which of the following points lies on x-axis?
Solution A point on the x-axis has y-coordinate 0. (2, 0) satisfies this.
9. Which point lies in the first quadrant?
Solution First quadrant has (+, +) coordinates. (3, 4) fits.
10. Distance between points (1, 2) and (4, 6) is:
Solution \(d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5\).
Section B: Assertion-Reason Questions
Directions:
(a) Both A and R are true and R is correct explanation of A.
(b) Both A and R are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
(a) Both A and R are true and R is correct explanation of A.
(b) Both A and R are true but R is not correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
11. Assertion (A): The point (0, 4) lies on y axis.
Reason (R): The x-coordinate on the point on y-axis is zero.
Reason (R): The x-coordinate on the point on y-axis is zero.
Solution Point (0, 4) has x=0, so it is on the y-axis. The Reason correctly states the property of y-axis points.
12. 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 formula is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
Reason (R): Distance formula is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
Solution Distance = 10. \(\sqrt{(10-2)^2 + (y+3)^2} = 10 \Rightarrow 64 + (y+3)^2 = 100 \Rightarrow (y+3)^2 = 36 \Rightarrow y+3 = \pm 6\). So \(y = 3\) or \(y = -9\). Assertion says y=6, which is incorrect. Reason is correct.
13. 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 (-5, 4) and (-2, 3) is 1:2.
Reason (R): The ratio in which the point (-3, k) divides the line segment joining (-5, 4) and (-2, 3) is 1:2.
Solution A is True (definition of midpoint). For R, let ratio be k:1. \(-3 = \frac{-2k - 5}{k+1} \Rightarrow -3k - 3 = -2k - 5 \Rightarrow -k = -2 \Rightarrow k = 2\). Ratio is 2:1. Reason says 1:2, so False.
14. Assertion (A): The point which divides the line joining A(1, 2) and B(-1, 1) internally in the ratio 1:2 is \((1/3, 5/3)\).
Reason (R): Section formula coordinates are \((\frac{m_1x_2+m_2x_1}{m_1+m_2}, \frac{m_1y_2+m_2y_1}{m_1+m_2})\).
Reason (R): Section formula coordinates are \((\frac{m_1x_2+m_2x_1}{m_1+m_2}, \frac{m_1y_2+m_2y_1}{m_1+m_2})\).
Solution \(x = \frac{1(-1)+2(1)}{3} = 1/3\). \(y = \frac{1(1)+2(2)}{3} = 5/3\). Assertion matches result. Reason is correct formula.
15. Assertion (A): The point on the X-axis which is equidistant from (-2, 3) and B(5, 4) is (2, 0).
Reason (R): The coordinates of the point P(x, y) which divides the line segment joining A and B in ratio \(m_1:m_2\) is given by section formula.
Reason (R): The coordinates of the point P(x, y) which divides the line segment joining A and B in ratio \(m_1:m_2\) is given by section formula.
Solution A: Let point be (x, 0). \((x+2)^2 + 9 = (x-5)^2 + 16 \Rightarrow x^2+4x+13 = x^2-10x+41 \Rightarrow 14x = 28 \Rightarrow x=2\). True.
R: True formula, but equidistance is based on distance formula, not section formula directly. Hence (b).
R: True formula, but equidistance is based on distance formula, not section formula directly. Hence (b).
16. Assertion (A): Ratio in which line \(3x + 4y = 7\) divides the line segment joining (1, 2) and (-2, 1) is 3:5.
Reason (R): The coordinates of the point P(x, y) dividing in ratio \(m_1:m_2\) is given by section formula.
Reason (R): The coordinates of the point P(x, y) dividing in ratio \(m_1:m_2\) is given by section formula.
Solution Ratio \(k = -\frac{ax_1+by_1+c}{ax_2+by_2+c}\). Line: \(3x+4y-7=0\).
\(k = -\frac{3(1)+4(2)-7}{3(-2)+4(1)-7} = -\frac{4}{-9} = 4/9\). Assertion says 3:5, False. Reason is True.
\(k = -\frac{3(1)+4(2)-7}{3(-2)+4(1)-7} = -\frac{4}{-9} = 4/9\). Assertion says 3:5, False. Reason is True.
17. Assertion (A): C is mid point of PQ, if P is (4, x), C is (y, -1) and Q is (-2, 4), then x and y respectively are -6 and 1.
Reason (R): The mid point of line segment joining \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
Reason (R): The mid point of line segment joining \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
Solution Midpoint \((\frac{4-2}{2}, \frac{x+4}{2}) = (1, \frac{x+4}{2})\).
Given C is \((y, -1)\). So \(y=1\) and \(\frac{x+4}{2} = -1 \Rightarrow x+4 = -2 \Rightarrow x = -6\). True. Reason explains it.
Given C is \((y, -1)\). So \(y=1\) and \(\frac{x+4}{2} = -1 \Rightarrow x+4 = -2 \Rightarrow x = -6\). True. Reason explains it.
18. Assertion (A): The point (-1, 6) divides the line segment joining (-3, 10) and (6, -8) in the ratio 2:7 internally.
Reason (R): Three points A, B and C are collinear if AB + BC = AC.
Reason (R): Three points A, B and C are collinear if AB + BC = AC.
Solution Using section formula for 2:7 with (-3,10) and (6,-8):
\(x = \frac{2(6)+7(-3)}{9} = \frac{12-21}{9} = -1\). \(y = \frac{2(-8)+7(10)}{9} = \frac{54}{9} = 6\). Point is (-1, 6). Assertion True. Reason is also True but unrelated.
\(x = \frac{2(6)+7(-3)}{9} = \frac{12-21}{9} = -1\). \(y = \frac{2(-8)+7(10)}{9} = \frac{54}{9} = 6\). Point is (-1, 6). Assertion True. Reason is also True but unrelated.
19. Assertion (A): The possible value of x for which the distance between A(x, -1) and B(5, 3) is 5 units are 2 and 8.
Reason (R): Distance formula is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
Reason (R): Distance formula is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
Solution \(\sqrt{(5-x)^2 + (3+1)^2} = 5 \Rightarrow (5-x)^2 + 16 = 25 \Rightarrow (5-x)^2 = 9 \Rightarrow 5-x = \pm 3\).
\(x = 2\) or \(x = 8\). Both correct. R is correct explanation.
\(x = 2\) or \(x = 8\). Both correct. R is correct explanation.
20. Assertion (A): If the points A(4, 3) and B(x, 5) lie on a circle with center O(2, 3), then value of x is 2.
Reason (R): The mid point of line segment joining P and Q is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
Reason (R): The mid point of line segment joining P and Q is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
Solution Radii equal: \(OA = OB \Rightarrow OA^2 = OB^2\).
\((4-2)^2 + (3-3)^2 = (x-2)^2 + (5-3)^2 \Rightarrow 4 = (x-2)^2 + 4 \Rightarrow (x-2)^2 = 0 \Rightarrow x=2\). True.
Reason is True but irrelevant to circle property.
\((4-2)^2 + (3-3)^2 = (x-2)^2 + (5-3)^2 \Rightarrow 4 = (x-2)^2 + 4 \Rightarrow (x-2)^2 = 0 \Rightarrow x=2\). True.
Reason is True but irrelevant to circle property.