Coordinate Geometry

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.

The question asks to identify which of the given coordinate pairs represents a point that lies on the y-axis.

The key concept is the definition of the coordinate axes. Any point on the y-axis has an x-coordinate of 0. Its coordinates are of the form (0, y).
Similarly, any point on the x-axis has a y-coordinate of 0, with coordinates of the form (x, 0).

We will examine the x-coordinate of each given point.
(A) (2, 0): The x-coordinate is 2. This point lies on the x-axis.
(B) (0, 2): The x-coordinate is 0. This point lies on the y-axis.
(C) (1, 5): The x-coordinate is 1. This point lies in the first quadrant.
(D) (5, 1): The x-coordinate is 5. This point lies in the first quadrant.

The point with an x-coordinate of 0 is (0, 2). Therefore, it is the point on the y-axis.