Question

The coordinates of two points of a line are (3, 1) and (5, 2).
Read the following statements.
(i) Slope of the line is 2
(ii) Slope of the line is (1)/(2)
(iii) (9, 4) is a point on this line
(iv) (4, 9) is a point on this line
Now choose the correct answer from those given below.

• A. (i) and (iii) are true
• B. (ii) and (iv) are true
• C. (ii) and (iii) are true
• D. (i) and (iv) are true

Hint:

Once you find the slope is 1/2, you can immediately eliminate options (A) and (D). Then you only need to check one of the points, (9,4) or (4,9), to decide between (B) and (C). This saves time in a multiple-choice setting.

Answer 1

The Correct Option is C

We are given two points on a line and four statements about the line's slope and other points on it. We need to identify which pair of statements is correct.

1. Slope of a line:
Given two points (x₁, y₁) and (x₂, y₂), the slope m is calculated as:
m = (y₂ - y₁)/(x₂ - x₁) 2. Equation of a line:
Using the point-slope form: y - y₁ = m(x - x₁).
3. Checking a point:
To check if a point lies on a line, substitute its coordinates into the line's equation. If the equation holds true, the point is on the line.

The given points are (x₁, y₁) = (3, 1) and (x₂, y₂) = (5, 2).

Evaluating statements (i) and (ii): Find the slope.
m = (2 - 1)/(5 - 3) = (1)/(2) The slope of the line is (1)/(2).
Therefore, Statement (i) is false and Statement (ii) is true.

Evaluating statements (iii) and (iv): Check the points.
First, let's find the equation of the line using point (3, 1) and slope m = (1)/(2).
y - 1 = (1)/(2)(x - 3) 2(y - 1) = x - 3 2y - 2 = x - 3 x - 2y - 1 = 0 This is the equation of the line.
Now, check point (9, 4) from Statement (iii):
Substitute x = 9 and y = 4 into the equation:
(9) - 2(4) - 1 = 9 - 8 - 1 = 0 Since 0 = 0, the equation holds true. So, Statement (iii) is true.
Check point (4, 9) from Statement (iv):
Substitute x = 4 and y = 9 into the equation:
(4) - 2(9) - 1 = 4 - 18 - 1 = -15 Since -15 ≠ 0, the equation does not hold true. So, Statement (iv) is false.

Statements (ii) and (iii) are true. Therefore, option (C) is the correct choice.