Let a,b, c and d be non-zero numbers. If the point of intersection of the lines `4ax + 2ay+c=0` and `5bx+2by +d=0` lies in the fourth quadrant and is equidistant from the two axes, then

Options

A. 2bc-3ad = 0
B. 2bc+3ad=0
C. 3bc-2ad=0
D. 3bc+2ad=0

Explanation

To solve the problem step-by-step, we will analyze the given lines and the conditions provided.

### Step 1: Identify the equations of the lines
We have two lines given by the equations:
1. \( 4ax + 2ay + c = 0 \)
2. \( 5bx + 2by + d = 0 \)

### Step 2: Determine the point of intersection
To find the point of intersection of the two lines, we will solve these equations simultaneously.

From the first equation, we can express \( y \) in terms of \( x \):
\[ 2ay = -4ax - c \implies y = -\frac{4a}{2a}x - \frac{c}{2a} = -2x - \frac{c}{2a} \]

Substituting this expression for \( y \) into the second equation:
\[ 5bx + 2b\left(-2x - \frac{c}{2a}\right) + d = 0 \]
This simplifies to:
\[ 5bx - 4bx - \frac{bc}{a} + d = 0 \implies (5b - 4b)x + d - \frac{bc}{a} = 0 \]
\[ bx + d - \frac{bc}{a} = 0 \implies bx = \frac{bc}{a} - d \implies x = \frac{bc}{ab} - \frac{d}{b} \]

### Step 3: Find \( y \) using \( x \)
Now substituting \( x \) back into the expression for \( y \):
\[ y = -2\left(\frac{bc}{ab} - \frac{d}{b}\right) - \frac{c}{2a} \]
This gives:
\[ y = -\frac{2bc}{ab} + \frac{2d}{b} - \frac{c}{2a} \]

### Step 4: Conditions for the point of intersection
The point of intersection lies in the fourth quadrant, which means \( x > 0 \) and \( y < 0 \). Additionally, it is equidistant from the axes, which implies \( |x| = |y| \).

### Step 5: Set up the equations
From the condition \( |x| = |y| \), we can set:
\[ \frac{bc}{ab} - \frac{d}{b} = -\left(-\frac{2bc}{ab} + \frac{2d}{b} - \frac{c}{2a}\right) \]

### Step 6: Simplify the equation
This leads to:
\[ \frac{bc}{ab} - \frac{d}{b} = \frac{2bc}{ab} - \frac{2d}{b} + \frac{c}{2a} \]
Rearranging gives:
\[ \frac{bc}{ab} + \frac{d}{b} - \frac{2bc}{ab} + \frac{2d}{b} - \frac{c}{2a} = 0 \]
This simplifies to:
\[ -\frac{bc}{ab} + \frac{3d}{b} - \frac{c}{2a} = 0 \]

### Step 7: Final relation
Multiplying through by \( 2ab \) to eliminate the denominators gives:
\[ -2c + 6bd - ac = 0 \implies 3bc - 2ad = 0 \]

### Conclusion
Thus, the relation between \( a, b, c, \) and \( d \) is:
\[ 3bc - 2ad = 0 \]