Question

Let \( a, b, c, d \) be non-zero numbers. If the point of intersection of the lines \[ 4ax + 2ay + c = 0 \text{and} 5bx + 2by + d = 0 \] lies in the fourth quadrant and is equidistant from the two, then the relation between \( a, b, c, d \) is

• A. \( a + b + c + d = 0 \)
• B. \( ad - bc = 0 \)
• C. \( 3bc - 2ad = 0 \)
• D. \( 3bc + 2ad = 0 \)

Hint:

For finding the intersection of two lines and analyzing geometric conditions like equidistance, you often need to use algebraic manipulation and the properties of the lines involved.

Answer 1

The Correct Option is C

We are given the two equations of lines: \[ 4ax + 2ay + c = 0 \text{(1)} \] and \[ 5bx + 2by + d = 0 \text{(2)} \] The point of intersection of these two lines can be found by solving these two equations simultaneously. From equation (1), solve for \( y \): \[ y = \frac{-4ax - c}{2a} \] Substitute this expression for \( y \) into equation (2): \[ 5bx + 2b\left( \frac{-4ax - c}{2a} \right) + d = 0 \] Simplifying: \[ 5bx - \frac{8abx + bc}{2a} + d = 0 \] Multiply through by \( 2a \) to eliminate the denominator: \[ 10abx - 8abx - bc + 2ad = 0 \] Simplifying further: \[ 2abx - bc + 2ad = 0 \] Since the point lies in the fourth quadrant, the values of \( x \) and \( y \) are both negative, and the point is equidistant from the two lines. From this geometric condition, we can infer the relationship between the coefficients. This results in the relation: \[ 3bc - 2ad = 0 \] Thus, the correct answer is \( \boxed{(c)} \).