The condition that the parabolas `y^2=4c(x-d)` and `y^2=4ax` have a common normal other than X-axis `(agt0,cgt0)` is

Options

A. `2alt2c+d`
B. `2clt2a+d`
C. `2dlt2a+c`
D. `2dlt2c+a`

Answer

To solve the problem of finding the condition that the parabolas \( y^2 = 4c(x - d) \) and \( y^2 = 4ax \) have a common normal other than the x-axis, we will follow these steps:

### Step 1: Write the normal equations for both parabolas

1. **For the parabola \( y^2 = 4ax \)**:
- The general normal equation in terms of slope \( m \) is given by:
\[ y = mx - 2am - am^3 \]

2. **For the parabola \( y^2 = 4c(x - d) \)**:
- The equation can be rewritten as \( y^2 = 4c(x - d) \). The normal equation becomes:
\[ y = m(x - d) - 2cm - cm^3 \]
- Simplifying this gives:
\[ y = mx - md - 2cm - cm^3 \]

### Step 2: Set the slopes equal

Since both parabolas have a common normal, the slopes of their normals must be equal. Thus, we equate the two normal equations:

\[ mx - 2am - am^3 = mx - md - 2cm - cm^3 \]

### Step 3: Eliminate \( mx \) and equate the constant terms

By eliminating \( mx \) from both sides, we get:

\[ -2am - am^3 = -md - 2cm - cm^3 \]

Rearranging gives:

\[ 2am + md + 2cm + am^3 + cm^3 = 0 \]

### Step 4: Factor out common terms

We can factor out \( m \) from the terms involving \( m \):

\[ m(2a + c + am^2 + cm^2) + md = 0 \]

### Step 5: Solve for \( m \)

Since \( m \neq 0 \) (we are looking for a normal other than the x-axis), we can divide by \( m \):

\[ 2a + c + am^2 + cm^2 + d = 0 \]

### Step 6: Rearranging the equation

Rearranging gives us:

\[ am^2 + cm^2 + 2a + c + d = 0 \]

### Step 7: Analyze the quadratic in \( m^2 \)

This is a quadratic equation in \( m^2 \):
\[ (a + c)m^2 + (2a + d + c) = 0 \]

For this quadratic to have real solutions, the discriminant must be non-negative:

\[ (2a + d + c)^2 - 4a(0) \geq 0 \]

### Step 8: Condition for common normal

Thus, we require:
\[ 2a + d + 2c \geq 0 \]

### Final Condition

After analyzing the conditions, we find that the condition for the parabolas to have a common normal other than the x-axis is:

\[ 2a < d + 2c \]