Equation of a Straight Line — Assertion-Reason Type Questions

y = -2 is the equation of a line.

Assertion (A): Its x-intercept is zero.

Reason (R): It does not intersect x-axis.

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 correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Line y = -2 is a line parallel to x-axis at a distance of 2 units from it.

For any point on x-axis the y-coordinate equals to zero, which is not the case with line y = -2.

So, it never intersects the x-axis.

So, reason (R) is true.

The x-intercept is the distance from the origin to the point where the line intersects the x-axis. Since this line never intersects the x-axis, so its x-intercept is undefined.

So, assertion (A) is false.

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

Hence, option 2 is the correct option.

The slope of a line passing through (-1, 0) is 1.

Assertion (A): Its x-intercept and y-intercept are equal.

Reason (R): It makes an isosceles triangle with the coordinate axes.

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

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

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

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

By point-slope form,

⇒ y - y₁ = m(x - x₁)

Equation of line passing through (-1, 0) and slope = 1 is :

⇒ y - 0 = 1[x - (-1)]

⇒ y = x + 1

⇒ x - y + 1 = 0

In order to find x-intercept, substitute y = 0 in the equation of line :

⇒ x - 0 + 1 = 0

⇒ x = -1

In order to find y-intercept, substitute x = 0 in the equation of line :

⇒ 0 - y + 1 = 0

⇒ y = 1

Thus, x-intercept and y-intercept are not equal.

So, assertion (A) is false.

The triangle are formed with the axes has vertices : (0, 0), (-1, 0) and (0, 1).

By distance formula,

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

$\text{Distance between (0, 0) and (-1, 0)} = \sqrt{(-1 - 0)^2 + (0 - 0)^2}\\[1em] = \sqrt{(-1)^2}\\[1em] = \sqrt{1}\\[1em] = 1 \\[1em] \text{Distance between (0, 0) and (0, 1)} = \sqrt{(0 - 0)^2 + (0 - 1)^2}\\[1em] = \sqrt{(-1)^2}\\[1em] = \sqrt{1}\\[1em] = 1$

Since,two sides are equal in length. Therefore, it is an isosceles triangle.

So, reason (R) is true.

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

Hence, option 2 is the correct option.

Given below are the equation of two lines:

y = 2x + 8 and y = $\dfrac{1}{2}x - 7$

Assertion (A): The two lines are perpendicular to each other.

Reason (R): Their slopes are reciprocal of each other.

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 correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Given lines,

⇒ y = 2x + 8 and y = $\dfrac{1}{2}x - 7$

Comparing above equations with y = mx + c we get,

Slope of 1st line = 2

Slope of 2nd line = $\dfrac{1}{2}$

The slopes of 1st line and 2nd line are reciprocal of each other.

So, reason (R) is true.

⇒ Slope of 1st line × Slope of 2nd line

⇒ 2 × $\dfrac{1}{2}$

⇒ 1.

Since, product ≠ -1, so lines are not perpendicular as product of slope of perpendicular lines = -1.

So, assertion (A) is false.

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

Hence, option 2 is the correct option.

l and m are two lines in the figure given below:

Assertion (A): They have equal y-intercept.

Reason (R): Their inclination is same.

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 correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

Both lines intersect at the same point on the y-axis. So, they have the same y-intercept.

So, assertion (A) is true.

Given,

x-intercept are equal but they are of different signs (one is positive, one is negative).

tan θ = $\dfrac{y}{x}$

So, the inclination of both the lines will be of opposite signs.

So, reason (R) is false.

Thus, Assertion (A) is true, but Reason (R) is false.

Hence, option 1 is the correct option.

Assertion (A): The line 3x + 3y + 5 = 0 crosses the x-axis at the point $\Big(0, - \dfrac{5}{3}\Big)$.

Reason (R): The ordinate of every point on x - axis is zero.

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 correct, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Answer

We know that,

The ordinate (y-coordinate) of every point on x-axis is zero.

So, reason (R) is true.

Substituting y = 0, in the equation of line 3x + 3y + 5 = 0, we get :

⇒ 3x + 3 x 0 + 5 = 0

⇒ 3x + 5 = 0

⇒ x = $-\dfrac{5}{3}$

Point of intersection = $\Big(-\dfrac{5}{3}, 0\Big)$

So, assertion (A) is false.

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

Hence, option 2 is the correct option.