Q.Find the distance between parallel lines

Concept: Distance between parallel lines — pick a point on one line and use the perpendicular distance formula to the other line.

(i) Lines: $15x + 8y - 34 = 0$ and $15x + 8y + 31 = 0$.

Take a point on the first line. For $x = 0$, $8y = 34 \Rightarrow y = \frac{17}{4}$. Point is $(0, \frac{17}{4})$.

Distance from this point to the second line:

(ii) Lines: $l(x + y) + p = 0$ and $l(x + y) - r = 0$.

Rewrite as $lx + ly + p = 0$ and $lx + ly - r = 0$. …

Why this formula works

The distance from a point to a line is the perpendicular distance. For parallel lines, every point on one line is the same perpendicular distance from the other line. So we can pick any convenient point on the first line, find its perpendicular distance to the second line, and that gives the distance between the lines.

The formula comes from taking a point on the first line (say where $x=0$, then $y = -C_1/B$) and plugging it into the distance-from-point-to-line formula for the second line. The $Ax_0 + By_0$ part cancels to $-C_1$, leaving $| -C_1 + C_2| = |C_1 - C_2|$ in the numerator.

Part (i): $15x + 8y - 34 = 0$ and $15x + 8y + 31 = 0$

Step 1: Identify $A$, $B$, $C_1$, $C_2$

Both lines have $A = 15$, $B = 8$. For the first line, $C_1 = -34$. For the second line, $C_2 = 31$.

Step 2: Apply the formula

Step 3: Simplify

$\sqrt{289} = 17$, so $d = \frac{65}{17}$.

--- …