Obtained the condition that the parabola `y^(2)=4b(x-c)andy^(2)=4ax` have a common normal other than x-axis `(agtbgt0)`.

Answer

To find the condition that the parabolas \( y^2 = 4b(x - c) \) and \( y^2 = 4ax \) have a common normal other than the x-axis, we can follow these steps:

### Step 1: Write the equations of the normals for both parabolas.
For the parabola \( y^2 = 4ax \), the equation of the normal at point \( (at^2, 2at) \) is given by:
\[ y = mx - 2am - at^3 \]
where \( m \) is the slope of the normal.

For the parabola \( y^2 = 4b(x - c) \), we can rewrite it as \( y^2 = 4bx - 4bc \). The equation of the normal at point \( (b\mu^2 + c, 2b\mu) \) is:
\[ y = \mu x - 2b\mu - b\mu^3 \]

### Step 2: Set the equations of the normals equal to each other.
Since we are looking for a common normal, we set the two equations equal:
\[ mx - 2am - at^3 = \mu x - 2b\mu - b\mu^3 \]

### Step 3: Rearranging the equation.
Rearranging gives us:
\[ (mx - \mu x) = (2b\mu - 2am) + (at^3 - b\mu^3) \]
This simplifies to:
\[ (m - \mu)x = 2b\mu - 2am + at^3 - b\mu^3 \]

### Step 4: Equate coefficients.
For the above equation to hold for all \( x \), the coefficients must be equal. Therefore, we have:
1. \( m - \mu = 0 \) (which implies \( m = \mu \))
2. \( 2b\mu - 2am + at^3 - b\mu^3 = 0 \)

### Step 5: Substitute \( m \) with \( \mu \).
Substituting \( m \) with \( \mu \) in the second equation gives:
\[ 2b\mu - 2a\mu + at^3 - b\mu^3 = 0 \]
This can be rearranged to:
\[ \mu(2b - 2a - b\mu^2) + at^3 = 0 \]

### Step 6: Analyze the equation.
From this equation, we can find the conditions under which it holds. We can express it as:
\[ \mu(2(b - a) - b\mu^2) + at^3 = 0 \]
This implies that for a common normal to exist, the discriminant of the quadratic in \( \mu \) must be non-negative.

### Step 7: Find the discriminant.
The discriminant \( D \) of the quadratic \( -b\mu^2 + 2(b - a)\mu + at^3 = 0 \) must be greater than or equal to zero:
\[ D = [2(b - a)]^2 - 4(-b)(at^3) \geq 0 \]
This leads to:
\[ 4(b - a)^2 + 4ab t^3 \geq 0 \]
Since \( a, b > 0 \), this condition will always hold.

### Step 8: Conclusion.
Thus, the condition for the parabolas \( y^2 = 4b(x - c) \) and \( y^2 = 4ax \) to have a common normal other than the x-axis is:
\[ 2a < 2b + c \]