If the point of intersection of the line `2ax + 4ay+c =0 and 7bx +3by-d=0` lies in the `4^(th)` quadrant and is equidistant from the two axes, where a,b,c and d are non-zero numbers, then ad:bc equals to

Options

A. `2:3`
B. `2:1`
C. `1:1`
D. `3:2`

Answer

To solve the problem step by step, we need to find the point of intersection of the two lines given and determine the ratio of \( ad:bc \).

### Step 1: Identify the lines
The equations of the lines are:
1. \( 2ax + 4ay + c = 0 \) (Line 1)
2. \( 7bx + 3by - d = 0 \) (Line 2)

### Step 2: Find the point of intersection
To find the point of intersection, we can solve the two equations simultaneously.

From Line 1:
\[ 2ax + 4ay + c = 0 \implies 4ay = -2ax - c \implies y = -\frac{2a}{4a}x - \frac{c}{4a} = -\frac{1}{2}x - \frac{c}{4a} \]

From Line 2:
\[ 7bx + 3by - d = 0 \implies 3by = -7bx + d \implies y = -\frac{7b}{3b}x + \frac{d}{3b} = -\frac{7}{3}x + \frac{d}{3b} \]

### Step 3: Set the equations for y equal to each other
Now we set the two expressions for \( y \) equal to find \( x \):
\[ -\frac{1}{2}x - \frac{c}{4a} = -\frac{7}{3}x + \frac{d}{3b} \]

### Step 4: Clear the fractions
To eliminate the fractions, we can multiply through by \( 12ab \):
\[ -6abx - 3bc = -28abx + 4ad \]

### Step 5: Rearranging the equation
Rearranging gives:
\[ (28ab - 6ab)x = 4ad + 3bc \]
\[ 22abx = 4ad + 3bc \]
\[ x = \frac{4ad + 3bc}{22ab} \]

### Step 6: Substitute x back to find y
Now substitute \( x \) back into one of the equations to find \( y \). Using the expression from Line 1:
\[ y = -\frac{1}{2}\left(\frac{4ad + 3bc}{22ab}\right) - \frac{c}{4a} \]

### Step 7: Equidistant from the axes
Since the point of intersection lies in the 4th quadrant and is equidistant from the axes, we have:
\[ y = -x \]
Thus, we can set:
\[ -\frac{1}{2}\left(\frac{4ad + 3bc}{22ab}\right) - \frac{c}{4a} = -\frac{4ad + 3bc}{22ab} \]

### Step 8: Solve for the relationship between a, b, c, d
From the above equation, we can derive:
\[ \frac{4ad + 3bc}{22ab} = \frac{c}{4a} \]
Cross-multiplying gives:
\[ 4ad + 3bc = \frac{22abc}{4a} \]
Simplifying this leads to:
\[ 4ad + 3bc = \frac{22bc}{4} \]

### Step 9: Find the ratio \( ad:bc \)
Rearranging gives:
\[ ad = \frac{2bc}{4} \implies \frac{ad}{bc} = \frac{2}{1} \]

Thus, the final ratio is:
\[ ad:bc = 2:1 \]

### Final Answer
The ratio \( ad:bc \) is \( 2:1 \).