Coordinate Geometry — Assertion-Reason Type Questions

Assertion (A): Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.

Reason (R): If a ≠ b, then (a, b) ≠ (b, a).

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Two ordered pairs (a, b) and (c, d) are equal if a = c and b = d.

This statement is the definition of equality for ordered pairs. For two ordered pairs to be considered equal, their corresponding elements must be equal.

So, (a, b) = (c, d) if and only if a = c and b = d.

∴ Assertion (A) is true.

If a ≠ b, then (a, b) ≠ (b, a).

For example, the ordered pair (2, 3) is not the same as (3,2). They represent different points in a coordinate plane.

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.

Assertion (A): The point P(3, -5) lies in quadrant II and the point Q(-2, -1) lies in quadrant III.

Reason (R): The signs of a point in I, II, III and IV quadrants are respectively (+, +), (-, +), (-, -) and (+, -).

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Point P(3, -5):

The x-coordinate is positive, and the y-coordinate is negative.

A point with a positive x-coordinate and a negative y-coordinate lies in quadrant IV.

Point Q(-2, -1):

The x-coordinate is negative, and the y-coordinate is negative.

A point with a negative x-coordinate and a negative y-coordinate lies in quadrant III.

∴ Assertion (A) is false.

Quadrant I: x-coordinate is positive, y-coordinate is positive (+, +).

Quadrant II: x-coordinate is negative, y-coordinate is positive (-, +).

Quadrant III: x-coordinate is negative, y-coordinate is negative (-, -).

Quadrant IV: x-coordinate is positive, y-coordinate is negative (+, -).

∴ Reason (R) is true.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): Two points A and B having coordinates (3, 3) and (-3, -3) respectively are joined. The line segment AB passes through the origin.

Reason (R): Origin is the point of intersection of the coordinate axes.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

When a point (x,y) is reflected across the origin (0, 0), its new coordinates become (−x, −y). This transformation is also known as point symmetry with respect to the origin.

Point A has coordinates (3, 3).

If we reflect point A(3, 3) across the origin (0, 0), the coordinates of its reflected image would be (−3, −3).

The coordinates of the given point B are exactly (−3, −3).

Since point B is the exact reflection of point A across the origin, it means that the origin (0, 0) lies precisely on the straight line segment connecting point A and point B.

Therefore, the line segment AB passes through the origin.

∴ Assertion (A) is true.

Origin is the point of intersection of the coordinate axes.

This is the definition of the origin in a Cartesian coordinate system. It's the point where the x-axis and y-axis intersect, and its coordinates are (0, 0).

∴ Reason (R) is true.

∴ Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.

Assertion (A): If the points A(2, 9), B(2, 5) and C(5, 5) are joined, then ΔABC is right angled.

Reason (R): If AC² = AB² + BC², then ∠B = 90°.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

Given, the points A(2, 9), B(2, 5) and C(5, 5).

By distance formula,

Distance between two points = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Given, points A(2, 9), B(2, 5) and C(5, 5).

$\text{Distance between AB }= \sqrt{(2 - 2)^2 + (5 - 9)^2} \\[1em] = \sqrt{0 + (-4)^2} \\[1em] = \sqrt{0 + 16} \\[1em] = \sqrt{16} \\[1em] = \text{4 units}. \\[1em] \text{Distance between BC } = \sqrt{(5 - 2)^2 + (5 - 5)^2} \\[1em] = \sqrt{3^2 + 0^2} \\[1em] = \sqrt{9 + 0} \\[1em] = \sqrt{9} \\[1em] = \text{3 units}. \\[1em] \text{Distance between AC } = \sqrt{(5 - 2)^2 + (5 - 9)^2} \\[1em] = \sqrt{3^2 + (-4)^2} \\[1em] = \sqrt{9 + 16} \\[1em] = \sqrt{25} \\[1em] = \text{5 units}.$

AC² = 5² = 25 units

AB² = 4² = 16 units

BC² = 3² = 9 units

Since, AC² = AB² + BC², thus it satisfies pythagoras theorem.

Thus, ABC is a right angled triangle at B.

Thus, Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): Point (0, 9) is a point on y-axis which is equidistant from points (6, 5) and (-4, 3).

Reason (R): Abscissa of a point on y-axis is 0.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

A point lies on the y-axis if and only if its x-coordinate (abscissa) is 0.

∴ Reason (R) is true.

Let P = (0, 9), A = (6, 5), and B = (-4, 3).

By formula,

Distance between two points = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$PA = \sqrt{(6 - 0)^2 + (5 - 9)^2}\\[1em] = \sqrt{6^2 + (-4)^2}\\[1em] = \sqrt{36 + 16}\\[1em] = \sqrt{52}. \\[1em] PB = \sqrt{(-4 - 0)^2 + (3 - 9)^2}\\[1em] = \sqrt{(-4)^2 + (-6)^2}\\[1em] = \sqrt{16 + 36}\\[1em] = \sqrt{52}$

Since PA = PB = $\sqrt{52}$.

​Therefore, the point (0, 9) is a point on y-axis which is equidistant from (6, 5) and (-4, 3).

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is not the correct reason (or explanation) for Assertion (A).

Hence, option 4 is the correct option.