If `a ne 0` and the line `2bx + 3cy + 4d = 0` passes through the points of intersection of the parabolas `y^2 = 4ax` and `x^2 = 4ay` then :

Options

A. `d^2+(2b+3c)^2=0`
B. `d^2+(3b+2c)^2=0`
C. `d^2+(2b-3c)^2=0`
D. `d^2+(3b-2c)^2=0`

Answer

To solve the problem step by step, we need to find the conditions under which the line \(2bx + 3cy + 4d = 0\) passes through the points of intersection of the parabolas \(y^2 = 4ax\) and \(x^2 = 4ay\).

### Step 1: Identify the Points of Intersection
We start with the two parabolas:
1. \(y^2 = 4ax\) (Parabola 1)
2. \(x^2 = 4ay\) (Parabola 2)

To find the points of intersection, we can express \(x\) from the first equation and substitute it into the second equation.

From \(y^2 = 4ax\), we can express \(x\) as:
\[ x = \frac{y^2}{4a} \]

Substituting \(x\) into the second parabola's equation:
\[ \left(\frac{y^2}{4a}\right)^2 = 4ay \]
\[ \frac{y^4}{16a^2} = 4ay \]
Multiplying both sides by \(16a^2\) to eliminate the fraction:
\[ y^4 = 64a^3y \]
Rearranging gives:
\[ y^4 - 64a^3y = 0 \]
Factoring out \(y\):
\[ y(y^3 - 64a^3) = 0 \]

This gives us:
1. \(y = 0\)
2. \(y^3 - 64a^3 = 0 \Rightarrow y = 4a\)

### Step 2: Find Corresponding \(x\) Values
Now we find the corresponding \(x\) values for the \(y\) values we found:
- For \(y = 0\):
\[ y^2 = 4ax \Rightarrow 0 = 4ax \Rightarrow x = 0 \]
- For \(y = 4a\):
\[ y^2 = 4ax \Rightarrow (4a)^2 = 4ax \Rightarrow 16a^2 = 4ax \Rightarrow x = 4a \]

Thus, the points of intersection are:
1. \(O(0, 0)\)
2. \(A(4a, 4a)\)

### Step 3: Substitute Points into the Line Equation
Now we check if the line \(2bx + 3cy + 4d = 0\) passes through these points.

1. For point \(O(0, 0)\):
\[ 2b(0) + 3c(0) + 4d = 0 \Rightarrow 4d = 0 \Rightarrow d = 0 \]

2. For point \(A(4a, 4a)\):
\[ 2b(4a) + 3c(4a) + 4d = 0 \Rightarrow 8ab + 12ac + 4d = 0 \]
Substituting \(d = 0\):
\[ 8ab + 12ac = 0 \]

### Step 4: Factor and Solve for Conditions
Factoring out \(4a\) (since \(a \neq 0\)):
\[ 4a(2b + 3c) = 0 \]
Since \(a \neq 0\), we have:
\[ 2b + 3c = 0 \]

### Conclusion
Thus, the condition for the line to pass through the points of intersection of the two parabolas is:
\[ 2b + 3c = 0 \]