✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 7: Coordinate Geometry
Assertion-Reason Question Bank
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Assertion (A)
Section formula for internal division in ratio \(m_1:m_2\) is \(\left(\frac{m_1x_2 + m_2x_1}{m_1+m_2}, \frac{m_1y_2 + m_2y_1}{m_1+m_2}\right)\).
Reason (R)
Mid-point formula is a special case of section formula with \(m_1 = m_2 = 1\).
Correct Option: (B)
Both A and R are true section formula facts.
Both A and R are true section formula facts.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The ratio in which y-axis divides segment joining \((5, -6)\) and \((-1, -4)\) is \(5:1\).
Reason (R)
The x-coordinate of point of division on y-axis is \(y\).
Correct Option: (C)
Assertion (A) is true (\(x = \frac{k(-1)+1(5)}{k+1} = 0 \implies -k+5=0 \implies k=5\), ratio \(5:1\)). Reason (R) is false because x-coordinate on y-axis is \(0\), not \(y\).
Assertion (A) is true (\(x = \frac{k(-1)+1(5)}{k+1} = 0 \implies -k+5=0 \implies k=5\), ratio \(5:1\)). Reason (R) is false because x-coordinate on y-axis is \(0\), not \(y\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The distance of point \(P(4, -3)\) from y-axis is \(-4\) units.
Reason (R)
Distance is a geometric measure and can never be negative; distance from y-axis is \(|x| = 4\) units.
Correct Option: (D)
Assertion (A) is false (distance is always non-negative \(+4\)). Reason (R) is true.
Assertion (A) is false (distance is always non-negative \(+4\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The centroid of triangle with vertices \((0,0), (3,0), (0,3)\) is \((3, 3)\).
Reason (R)
Centroid is \(G = \left(\frac{0+3+0}{3}, \frac{0+0+3}{3}\right) = (1, 1)\).
Correct Option: (D)
Assertion (A) is false (it is \((1,1)\)). Reason (R) is true.
Assertion (A) is false (it is \((1,1)\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The distance between \((0, 5)\) and \((-5, 0)\) is \(5\sqrt{2}\) units.
Reason (R)
Distance formula is \(d = \sqrt{(x_2+x_1)^2 + (y_2+y_1)^2}\).
Correct Option: (C)
Assertion (A) is true (\(d = \sqrt{25+25} = \sqrt{50} = 5\sqrt{2}\)). Reason (R) is false because distance formula uses subtraction \((x_2-x_1)^2\), not addition.
Assertion (A) is true (\(d = \sqrt{25+25} = \sqrt{50} = 5\sqrt{2}\)). Reason (R) is false because distance formula uses subtraction \((x_2-x_1)^2\), not addition.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The centroid of triangle with vertices \(A(1, 4), B(2, -2), C(3, 1)\) is \((2, 1)\).
Reason (R)
Centroid formula is \(G = \left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right) = \left(\frac{1+2+3}{3}, \frac{4-2+1}{3}\right) = (2, 1)\).
Correct Option: (A)
Centroid \(G = ((1+2+3)/3, (4-2+1)/3) = (6/3, 3/3) = (2, 1)\). Reason explains Assertion.
Centroid \(G = ((1+2+3)/3, (4-2+1)/3) = (6/3, 3/3) = (2, 1)\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The area of triangle with vertices \((0,0), (a,0), (0,b)\) is \(\frac{1}{2}|ab|\).
Reason (R)
The triangle formed by axes and line \(x/a + y/b = 1\) is right-angled at origin.
Correct Option: (B)
Both A and R are true. Area \(= (1/2) \times \text{base} \times \text{height} = (1/2)ab\).
Both A and R are true. Area \(= (1/2) \times \text{base} \times \text{height} = (1/2)ab\).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The points \(A(1, 2), B(2, 4)\) and \(C(3, 6)\) are collinear.
Reason (R)
\(AB = \sqrt{5}, BC = \sqrt{5}, AC = 2\sqrt{5}\), satisfying \(AB + BC = AC\).
Correct Option: (A)
By distance formula: \(AB = \sqrt{5}, BC = \sqrt{5}, AC = 2\sqrt{5}\). Since \(AB + BC = AC\), points are collinear. Reason explains Assertion.
By distance formula: \(AB = \sqrt{5}, BC = \sqrt{5}, AC = 2\sqrt{5}\). Since \(AB + BC = AC\), points are collinear. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The x-axis divides the line segment joining \((2, -3)\) and \((5, 6)\) in ratio \(2:1\).
Reason (R)
Using \(y = \frac{k(6) + 1(-3)}{k+1} = 0 \implies 6k - 3 = 0 \implies k = 1/2\), ratio is \(1:2\).
Correct Option: (D)
Assertion (A) is false (ratio is \(1:2\), not \(2:1\)). Reason (R) is true.
Assertion (A) is false (ratio is \(1:2\), not \(2:1\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The point \(P(0, -2)\) lies on the y-axis.
Reason (R)
The point \(Q(3, 0)\) lies on the x-axis.
Correct Option: (B)
Both A and R are true facts about points on coordinate axes.
Both A and R are true facts about points on coordinate axes.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The perimeter of triangle formed by \((0,4), (0,0), (3,0)\) is \(12\) units.
Reason (R)
The sides of right triangle are \(3, 4\) and hypotenuse \(\sqrt{3^2+4^2} = 5\).
Correct Option: (B)
Both A and R are true (\(3+4+5=12\)).
Both A and R are true (\(3+4+5=12\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
If \(P(x, y)\) is equidistant from \(A(5, 1)\) and \(B(-1, 5)\), then \(x = y\).
Reason (R)
Equidistant condition \(PA^2 = PB^2 \implies (x-5)^2 + (y-1)^2 = (x+1)^2 + (y-5)^2 \implies 3x = 2y\).
Correct Option: (D)
Assertion (A) is false (\(3x = 2y\), not \(x = y\)). Reason (R) is true.
Assertion (A) is false (\(3x = 2y\), not \(x = y\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The point \((3, 0)\) lies on the x-axis.
Reason (R)
The y-coordinate of every point on the x-axis is zero.
Correct Option: (B)
Both A and R are true coordinate geometry definitions.
Both A and R are true coordinate geometry definitions.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The origin \((0, 0)\) is the mid-point of line segment joining \((2, -3)\) and \((-2, 3)\).
Reason (R)
Mid-point formula is \(\left(\frac{x_1-x_2}{2}, \frac{y_1-y_2}{2}\right)\).
Correct Option: (C)
Assertion (A) is true (\(((2-2)/2, (-3+3)/2) = (0,0)\)). Reason (R) is false because formula is \((x_1+x_2)/2\), not \((x_1-x_2)/2\).
Assertion (A) is true (\(((2-2)/2, (-3+3)/2) = (0,0)\)). Reason (R) is false because formula is \((x_1+x_2)/2\), not \((x_1-x_2)/2\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The point \((2, -3)\) lies in the second quadrant.
Reason (R)
Points in the fourth quadrant have positive x-coordinate and negative y-coordinate \((+, -)\).
Correct Option: (D)
Assertion (A) is false (\((2,-3)\) lies in 4th quadrant). Reason (R) is true.
Assertion (A) is false (\((2,-3)\) lies in 4th quadrant). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If \(A(1, 2), B(4, y), C(x, 6)\) and \(D(3, 5)\) are vertices of a parallelogram taken in order, then \(x = 6\) and \(y = 3\).
Reason (R)
Diagonals of a parallelogram bisect each other.
Correct Option: (A)
Both A and R are true and R provides the property used (\(((1+x)/2, (2+6)/2) = ((4+3)/2, (y+5)/2) \implies x=6, y=3\)). Reason explains Assertion.
Both A and R are true and R provides the property used (\(((1+x)/2, (2+6)/2) = ((4+3)/2, (y+5)/2) \implies x=6, y=3\)). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The distance of point \(P(3, -4)\) from origin \((0, 0)\) is \(5\) units.
Reason (R)
The distance of a point \((x, y)\) from origin is \(\sqrt{x^2 + y^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
Correct Option: (A)
Formula \(d = \sqrt{x^2+y^2}\) gives \(\sqrt{9+16} = 5\) units. Reason gives exact calculation and explains Assertion.
Formula \(d = \sqrt{x^2+y^2}\) gives \(\sqrt{9+16} = 5\) units. Reason gives exact calculation and explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The mid-point of line segment joining \(A(2, -3)\) and \(B(6, 7)\) is \((4, 2)\).
Reason (R)
The mid-point formula is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{2+6}{2}, \frac{-3+7}{2}\right) = (4, 2)\).
Correct Option: (A)
Midpoint formula \(((2+6)/2, (-3+7)/2) = (4, 2)\). Reason gives exact formula and explains Assertion.
Midpoint formula \(((2+6)/2, (-3+7)/2) = (4, 2)\). Reason gives exact formula and explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The distance of point \(P(-3, 5)\) from x-axis is \(5\) units.
Reason (R)
The distance of a point from x-axis is its x-coordinate.
Correct Option: (C)
Assertion (A) is true (distance from x-axis = \(|y| = 5\)). Reason (R) is false because distance from x-axis is the absolute value of its y-coordinate, not x-coordinate.
Assertion (A) is true (distance from x-axis = \(|y| = 5\)). Reason (R) is false because distance from x-axis is the absolute value of its y-coordinate, not x-coordinate.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The point \(P(2, 3)\) divides line segment joining \(A(1, 1)\) and \(B(3, 5)\) in ratio \(1:1\).
Reason (R)
\(P\) is the mid-point of \(AB\), and mid-point divides segment in ratio \(1:1\).
Correct Option: (A)
Midpoint of \(A(1,1)\) and \(B(3,5)\) is \(((1+3)/2, (1+5)/2) = (2, 3)\), which divides in \(1:1\) ratio. Reason explains Assertion.
Midpoint of \(A(1,1)\) and \(B(3,5)\) is \(((1+3)/2, (1+5)/2) = (2, 3)\), which divides in \(1:1\) ratio. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The line \(y = 3\) is parallel to the x-axis.
Reason (R)
Every point on line \(y = 3\) has a y-coordinate equal to \(3\).
Correct Option: (B)
Both A and R are true geometric properties.
Both A and R are true geometric properties.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The distance between \(P(2, 3)\) and \(Q(2, 3)\) is \(1\) unit.
Reason (R)
The distance between two identical points is \(0\) units.
Correct Option: (D)
Assertion (A) is false (distance is \(0\)). Reason (R) is true.
Assertion (A) is false (distance is \(0\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The point \((0, 4)\) lies on the y-axis.
Reason (R)
The x-coordinate of every point lying on the y-axis is always zero.
Correct Option: (A)
On the y-axis, \(x=0\). Point \((0, 4)\) has \(x=0\), so it lies on y-axis. Reason explains Assertion.
On the y-axis, \(x=0\). Point \((0, 4)\) has \(x=0\), so it lies on y-axis. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The distance between \(A(a, b)\) and \(B(-a, -b)\) is \(2\sqrt{a^2 + b^2}\).
Reason (R)
By distance formula, \(AB = \sqrt{(-a - a)^2 + (-b - b)^2} = \sqrt{(-2a)^2 + (-2b)^2} = \sqrt{4a^2 + 4b^2} = 2\sqrt{a^2 + b^2}\).
Correct Option: (A)
Applying distance formula yields \(2\sqrt{a^2+b^2}\). Reason explains Assertion.
Applying distance formula yields \(2\sqrt{a^2+b^2}\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
The mid-point of segment joining \((3, 4)\) and \((5, 6)\) is \((8, 10)\).
Reason (R)
The mid-point coordinates are \(\left(\frac{3+5}{2}, \frac{4+6}{2}\right) = (4, 5)\).
Correct Option: (D)
Assertion (A) is false (it is \((4,5)\), not \((8,10)\)). Reason (R) is true.
Assertion (A) is false (it is \((4,5)\), not \((8,10)\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The distance of point \(P(-6, 8)\) from origin is \(10\) units.
Reason (R)
Distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
Correct Option: (B)
Both A and R are true. \(d = \sqrt{36+64} = 10\). R is general distance formula.
Both A and R are true. \(d = \sqrt{36+64} = 10\). R is general distance formula.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
The point \(A(-4, 0), B(4, 0), C(0, 3)\) are vertices of an isosceles triangle.
Reason (R)
Side lengths are \(AB = 8, BC = 5, AC = 6\).
Correct Option: (C)
Assertion (A) is true (\(AB = 8, BC = \sqrt{16+9}=5, AC = \sqrt{16+9}=5\), isosceles). Reason (R) is false because \(AC = 5\), not \(6\).
Assertion (A) is true (\(AB = 8, BC = \sqrt{16+9}=5, AC = \sqrt{16+9}=5\), isosceles). Reason (R) is false because \(AC = 5\), not \(6\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The points \((0, 0), (3, 0), (0, 4)\) form an equilateral triangle.
Reason (R)
Side lengths are \(3, 4, 5\), which form a right-angled scalene triangle.
Correct Option: (D)
Assertion (A) is false (it is right-angled scalene). Reason (R) is true.
Assertion (A) is false (it is right-angled scalene). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The points \((1, 7), (4, 2), (-1, -1), (-4, 4)\) are vertices of a square.
Reason (R)
All sides of a square are equal and diagonals are perpendicular but unequal.
Correct Option: (C)
Assertion (A) is true (sides \(= \sqrt{34}\), diagonals \(= \sqrt{68}\)). Reason (R) is false because diagonals of a square are EQUAL, not unequal.
Assertion (A) is true (sides \(= \sqrt{34}\), diagonals \(= \sqrt{68}\)). Reason (R) is false because diagonals of a square are EQUAL, not unequal.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
The point \(P(2, 3)\) is equidistant from \(A(2, 1)\) and \(B(2, 5)\).
Reason (R)
Distance \(PA = 3\) units and \(PB = 3\) units.
Correct Option: (C)
Assertion (A) is true (\(PA = \sqrt{0+4}=2, PB=\sqrt{0+4}=2\), so \(PA=PB=2\)). Reason (R) is false because distance is \(2\) units, not \(3\) units.
Assertion (A) is true (\(PA = \sqrt{0+4}=2, PB=\sqrt{0+4}=2\), so \(PA=PB=2\)). Reason (R) is false because distance is \(2\) units, not \(3\) units.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.