4.3 Q-23

Question Statement

Find the distance between the given parallel lines and sketch the lines. Also, find an equation of the parallel line lying midway between them.

(a) $3x - ay + 3 = 0$ and $3x - 4y + 7 = 0$

(b) $12x + 5y - 6 = 0$ and $12x + 5y + 13 = 0$

(c) $x + 2y - 5 = 0$ and $2x + 4y = 1$

---

Background and Explanation

• The distance between two parallel lines can be found using the formula:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

• The equation of a line passing through the midpoint of two points can be derived using the point-slope form of the line’s equation.

• A key concept is to first find the points where the lines intersect the y-axis by setting $x = 0$ and solving for $y$. Then, calculate the midpoint and use the slope to find the equation of the line between them.

---

Solution

Part (a)

Given lines:

1. $3x - 4y + 7 = 0$
2. $3x - 4y + 3 = 0$

• Step 1: Find the points where the lines intersect the y-axis (set $x = 0$):

For the first line:

$$0 - 4y + 7 = 0 \quad \Rightarrow \quad y = \frac{7}{4}$$

Hence, the point is $(0, \frac{7}{4})$.

For the second line:

$$0 - 4y + 3 = 0 \quad \Rightarrow \quad y = \frac{3}{4}$$

Hence, the point is $(0, \frac{3}{4})$.

• Step 2: Calculate the distance between the two points using the distance formula:

$$d = \frac{|(3)(0) + (-4)(\frac{3}{4}) + 7|}{\sqrt{(3)^2 + (-4)^2}} = \frac{|0 - 3 + 7|}{\sqrt{9 + 16}} = \frac{4}{5}$$

• Step 3: Find the midpoint of the points $(0, \frac{7}{4})$ and $(0, \frac{3}{4})$:

$$\text{Midpoint} = \left( \frac{0 + 0}{2}, \frac{\frac{7}{4} + \frac{3}{4}}{2} \right) = \left( 0, \frac{5}{4} \right)$$

• Step 4: Find the equation of the parallel line passing through the midpoint with the same slope as the given lines. The slope is $m = \frac{3}{4}$ (since the lines are parallel).

The equation of the line through $(0, \frac{5}{4})$ with slope $\frac{3}{4}$ is:

$$y - \frac{5}{4} = \frac{3}{4}(x - 0) \quad \Rightarrow \quad 3x - 7y + 5 = 0$$

---

Part (b)

Given lines:

1. $12x + 5y - 6 = 0$
2. $12x + 5y + 13 = 0$

• Step 1: Find the points where the lines intersect the y-axis (set $x = 0$):

For the first line:

$$0 + 5y - 6 = 0 \quad \Rightarrow \quad y = \frac{6}{5}$$

Hence, the point is $(0, \frac{6}{5})$.

For the second line:

$$0 + 5y + 13 = 0 \quad \Rightarrow \quad y = -\frac{13}{5}$$

Hence, the point is $(0, -\frac{13}{5})$.

• Step 2: Calculate the distance between the two points:

$$d = \frac{|(12)(0) + (5)(\frac{6}{5}) + 13|}{\sqrt{(12)^2 + (5)^2}} = \frac{|0 + 6 + 13|}{\sqrt{144 + 25}} = \frac{19}{13}$$

• Step 3: Find the midpoint of the points $(0, \frac{6}{5})$ and $(0, -\frac{13}{5})$:

$$\text{Midpoint} = \left( \frac{0 + 0}{2}, \frac{\frac{6}{5} + (-\frac{13}{5})}{2} \right) = \left( 0, -\frac{7}{10} \right)$$

• Step 4: Find the equation of the parallel line passing through the midpoint. The slope is $m = \frac{12}{5}$ (same as the given lines).

The equation of the line through $(0, -\frac{7}{10})$ with slope $\frac{12}{5}$ is:

$$y + \frac{7}{10} = \frac{12}{5}(x - 0) \quad \Rightarrow \quad 24x + 10y + 7 = 0$$

---

Part (c)

Given lines:

1. $x + 2y - 5 = 0$
2. $2x + 4y = 1$

• Step 1: Find the points where the lines intersect the y-axis (set $x = 0$):

For the first line:

$$0 + 2y - 5 = 0 \quad \Rightarrow \quad y = \frac{5}{2}$$

Hence, the point is $(0, \frac{5}{2})$.

For the second line:

$$0 + 4y = 1 \quad \Rightarrow \quad y = \frac{1}{4}$$

Hence, the point is $(0, \frac{1}{4})$.

• Step 2: Calculate the distance between the two points:

$$d = \frac{|(2)(0) + (4)(\frac{5}{2}) + (-1)|}{\sqrt{(2)^2 + (4)^2}} = \frac{|0 + 10 - 1|}{\sqrt{4 + 16}} = \frac{9}{2\sqrt{2}}$$

• Step 3: Find the midpoint of the points $(0, \frac{5}{2})$ and $(0, \frac{1}{4})$:

$$\text{Midpoint} = \left( \frac{0 + 0}{2}, \frac{\frac{5}{2} + \frac{1}{4}}{2} \right) = \left( 0, \frac{11}{8} \right)$$

• Step 4: Find the equation of the parallel line passing through the midpoint. The slope is $m = \frac{1}{2}$ (same as the given lines).

The equation of the line through $(0, \frac{11}{8})$ with slope $\frac{1}{2}$ is:

$$y - \frac{11}{8} = \frac{1}{2}(x - 0) \quad \Rightarrow \quad 2x - 7y + 5 = 0$$

---

Key Formulas or Methods Used

• Distance between two parallel lines:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

• Equation of a line using point-slope form:

$$y - y_1 = m(x - x_1)$$

---

Summary of Steps

1. Find the intersection points of the lines with the y-axis by setting $x = 0$.
2. Calculate the distance between the two points.
3. Find the midpoint of the points.
4. Use the midpoint and the slope of the lines to find the equation of the parallel line passing through the midpoint.