Question

The point (a, b) is the foot of the perpendicular drawn from the point (3, 1) to the line x + 3y + 4 = 0. If (p, q) is the image of (a, b) with respect to the line 3x - 4y + 11 = 0, then $\frac{p}{a} + \frac{q}{b} = $

• A. \(-3\)
• B. \(-5\)
• C. \(3\)
• D. \(7\)

Hint:

Remember the formulas for foot of the perpendicular and image of a point with respect to a line.

Answer 1

The Correct Option is B

Step 1: Find the foot of the perpendicular (a, b).
Let the point be P(3, 1) and the line be L: x + 3y + 4 = 0.
The slope of the line L is m_1 = -\(\frac{1}{3}\).
The slope of the line perpendicular to L is m_2 = -\(\frac{1}{m_1}\) = 3.
The equation of the line passing through P(3, 1) and perpendicular to L is:
y - 1 = 3(x - 3)
y - 1 = 3x - 9
3x - y - 8 = 0
To find (a, b), solve the equations x + 3y + 4 = 0 and 3x - y - 8 = 0.
From x + 3y + 4 = 0, we get x = -3y - 4.
Substitute in 3x - y - 8 = 0:
3(-3y - 4) - y - 8 = 0
-9y - 12 - y - 8 = 0
-10y - 20 = 0
y = -2
x = -3(-2) - 4 = 6 - 4 = 2
So, (a, b) = (2, -2).

Step 2: Find the image (p, q) of (a, b) with respect to the line 3x - 4y + 11 = 0.
Let the line be M: 3x - 4y + 11 = 0.
The midpoint of (a, b) and (p, q) lies on the line M.
Midpoint = \(\left(\frac{p+2}{2}, \frac{q-2}{2}\right)\)
Substitute in M:
3\(\left(\frac{p+2}{2}\right)\) - 4\(\left(\frac{q-2}{2}\right)\) + 11 = 0
3(p+2) - 4(q-2) + 22 = 0
3p + 6 - 4q + 8 + 22 = 0
3p - 4q + 36 = 0 ...(1)
The line joining (a, b) and (p, q) is perpendicular to M.
Slope of M = \(\frac{3}{4}\)
Slope of the line joining (a, b) and (p, q) = \(\frac{q+2}{p-2} = -\frac{4}{3}\)
3(q+2) = -4(p-2)
3q + 6 = -4p + 8
4p + 3q - 2 = 0 ...(2)
Solve (1) and (2):
From (2), 3q = 2 - 4p, so q = \(\frac{2-4p}{3}\).
Substitute in (1):
3p - 4\(\left(\frac{2-4p}{3}\right)\) + 36 = 0
9p - 4(2-4p) + 108 = 0
9p - 8 + 16p + 108 = 0
25p + 100 = 0
p = -4
q = \(\frac{2 - 4(-4)}{3} = \frac{2 + 16}{3} = \frac{18}{3} = 6\)
So, (p, q) = (-4, 6).

Step 3: Calculate \(\frac{p}{a} + \frac{q}{b}\).
\(\frac{p}{a} + \frac{q}{b} = \frac{-4}{2} + \frac{6}{-2} = -2 - 3 = -5\)
Therefore, \(\frac{p}{a} + \frac{q}{b} = -5\).