If `y^2=4b(x-c) and y^2 =8ax` having common normal then `(a,b,c)` is (a) `(1/2,2,0)` (b) `(1,1,3)` (c) `(1,1,1)` (d) `(1,3,2)`

Options

A. (1, 1, 0)
B. `((1)/(2), 2, 3)`
C. `((1)/(2), 2, 0)`
D. (1,1,3)

Answer

To solve the problem, we need to find the values of \( (a, b, c) \) such that the parabolas given by the equations \( y^2 = 4b(x - c) \) and \( y^2 = 8ax \) have a common normal.

### Step 1: Identify the equations of the parabolas
The first parabola is given by:
\[ y^2 = 4b(x - c) \]
This can be rewritten in standard form as:
\[ y^2 = 4b x - 4bc \]

The second parabola is:
\[ y^2 = 8ax \]

### Step 2: Find the slope of the normal for each parabola
For the parabola \( y^2 = 8ax \), the slope of the normal at a point \( (x_1, y_1) \) is given by:
\[ y = mx - 4am - 2am^3 \]
where \( m \) is the slope of the tangent at the point.

For the parabola \( y^2 = 4b(x - c) \), the equation of the normal at a point \( (x_2, y_2) \) is:
\[ y = mx - c - 2bm - bm^3 \]

### Step 3: Set the equations of the normals equal
Since the two parabolas have a common normal, we can equate the two normal equations:
\[ mx - 4am - 2am^3 = mx - c - 2bm - bm^3 \]

### Step 4: Simplify the equation
Cancelling \( mx \) from both sides, we get:
\[ -4am - 2am^3 = -c - 2bm - bm^3 \]
Rearranging gives:
\[ 4am + 2am^3 - 2bm - bm^3 + c = 0 \]

### Step 5: Factor and analyze the equation
We can factor out \( m \):
\[ m(4a + 2am^2 - 2b - bm^2) + c = 0 \]
This implies that for the equation to hold for all \( m \), the coefficients of \( m \) must equal zero. Thus, we have:
1. \( 4a - 2b + c = 0 \)
2. \( 2a - b = 0 \)

### Step 6: Solve the equations
From the second equation, we can express \( b \) in terms of \( a \):
\[ b = 2a \]

Substituting \( b = 2a \) into the first equation:
\[ 4a - 2(2a) + c = 0 \implies 4a - 4a + c = 0 \implies c = 0 \]

### Step 7: Find the values of \( a, b, c \)
Using \( b = 2a \) and \( c = 0 \):
- If we let \( a = 1 \), then \( b = 2 \times 1 = 2 \) and \( c = 0 \).
Thus, \( (a, b, c) = (1, 2, 0) \).

### Step 8: Check the options
The options provided are:
(a) \( (1/2, 2, 0) \)
(b) \( (1, 1, 3) \)
(c) \( (1, 1, 1) \)
(d) \( (1, 3, 2) \)

None of the options match \( (1, 2, 0) \). Let's check if \( (1, 1, 3) \) satisfies the conditions:
- If \( a = 1 \), then \( b = 2 \times 1 = 2 \) and \( c = 0 \) do not satisfy any of the options.

### Conclusion
Upon checking the values, the correct answer based on the analysis is not listed in the options provided. However, if we consider the conditions and the derived equations, we can conclude that the values \( (1, 2, 0) \) are derived correctly.