Find the equations of the lines joining the points
(i) A(1, 1) and B(2, 3) (ii) L(a, b) and M(b, a) (iii) P(3, 3) and Q(7, 6)
What is the length of the portion of the line in (c) intercepted between the axes of co-ordinates?

Answer

To find the equations of the lines joining the given points, we will use the slope-point form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

where \( m \) is the slope of the line, and \( (x_1, y_1) \) is a point on the line.

### Step-by-Step Solution:

#### (i) Points A(1, 1) and B(2, 3)

1. **Find the slope (m)**:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2 \]

2. **Use the slope-point form**:
Using point A(1, 1):
\[ y - 1 = 2(x - 1) \]

3. **Simplify the equation**:
\[ y - 1 = 2x - 2 \implies 2x - y - 1 = 0 \]

So, the equation of the line joining A and B is:
\[ 2x - y - 1 = 0 \]

#### (ii) Points L(a, b) and M(b, a)

1. **Find the slope (m)**:
\[ m = \frac{a - b}{b - a} = -1 \]

2. **Use the slope-point form**:
Using point L(a, b):
\[ y - b = -1(x - a) \]

3. **Simplify the equation**:
\[ y - b = -x + a \implies x + y - a - b = 0 \]

So, the equation of the line joining L and M is:
\[ x + y - a - b = 0 \]

#### (iii) Points P(3, 3) and Q(7, 6)

1. **Find the slope (m)**:
\[ m = \frac{6 - 3}{7 - 3} = \frac{3}{4} \]

2. **Use the slope-point form**:
Using point P(3, 3):
\[ y - 3 = \frac{3}{4}(x - 3) \]

3. **Simplify the equation**:
\[ 4(y - 3) = 3(x - 3) \implies 4y - 12 = 3x - 9 \implies 3x - 4y + 3 = 0 \]

So, the equation of the line joining P and Q is:
\[ 3x - 4y + 3 = 0 \]

### Length of the Portion of the Line in (iii) Intercepted Between the Axes

1. **Find the x-intercept**:
Set \( y = 0 \):
\[ 3x + 3 = 0 \implies x = -1 \]
So, the x-intercept is (-1, 0).

2. **Find the y-intercept**:
Set \( x = 0 \):
\[ 3(0) - 4y + 3 = 0 \implies -4y + 3 = 0 \implies y = \frac{3}{4} \]
So, the y-intercept is (0, 3/4).

3. **Calculate the length of the line segment between the intercepts**:
Using the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (-1, 0) \) and \( (x_2, y_2) = (0, \frac{3}{4}) \):
\[ \text{Distance} = \sqrt{(0 - (-1))^2 + \left(\frac{3}{4} - 0\right)^2} = \sqrt{(1)^2 + \left(\frac{3}{4}\right)^2} = \sqrt{1 + \frac{9}{16}} = \sqrt{\frac{16}{16} + \frac{9}{16}} = \sqrt{\frac{25}{16}} = \frac{5}{4} \]

So, the length of the portion of the line intercepted between the axes is:
\[ \frac{5}{4} \]