Consider parabola `P_(1)-=y=x^(2)` and `P_(2)-=y^(2)=-8x` and the line `L-=lx+my+n=0`. Which of the following holds true (a point `(alpha,beta)` is called rational point if `alpha` and `beta` are rational)

Options

A. a. If `l,m,n` are odd integers then the line `L` can not intersect parabola `P_(1)` in a rational point.
B. b. Line `L` will be tangent to `P_(1)` if `m,l/2,n` are in G.P.
C. c. If line `L` is common tangent to `P_(1)` and `P_(2)` then `l+m+n=0`
D. d. If line `L` is common chord of `P_(1)` and `P_(2)` then `l-2m+n=0`

Explanation

To solve the problem, we need to analyze the given parabolas and the line to determine the relationships between them. Let's break down the solution step by step.

### Step 1: Define the Parabolas and the Line
We have two parabolas:
1. \( P_1: y = x^2 \)
2. \( P_2: y^2 = -8x \)

And a line defined by:
\[ L: lx + my + n = 0 \]

### Step 2: Find the Condition for Tangency
To find the condition under which the line \( L \) is tangent to the parabola \( P_1 \), we substitute \( y = x^2 \) into the equation of the line:
\[ lx + m(x^2) + n = 0 \]
This can be rearranged to:
\[ mx^2 + lx + n = 0 \]

For the line to be tangent to the parabola, this quadratic equation must have exactly one solution, which occurs when the discriminant is zero:
\[ D = b^2 - 4ac = l^2 - 4mn = 0 \]
Thus, we have:
\[ l^2 = 4mn \]
This implies that \( l, m, n \) are in a geometric progression (GP).

### Step 3: Analyze Rational Points
A point \( (\alpha, \beta) \) is called a rational point if both \( \alpha \) and \( \beta \) are rational numbers.

If \( l, m, n \) are odd integers, we need to check if the line can intersect the parabola \( P_1 \) at a rational point.

Assuming \( x = \frac{p}{q} \) (where \( p \) and \( q \) are coprime integers), substituting into the line equation gives:
\[ L\left(\frac{p}{q}\right) + m\left(\frac{p}{q}\right)^2 + n = 0 \]
This leads to:
\[ mp^2 + Lpq + nq^2 = 0 \]

If \( p \) and \( q \) are both odd, then \( mp^2 \) (odd), \( Lpq \) (odd), and \( nq^2 \) (odd) cannot sum to zero, as the sum of three odd numbers is odd. Therefore, the line cannot intersect the parabola at a rational point.

### Step 4: Common Tangent Condition
For the line \( L \) to be a common tangent to both parabolas \( P_1 \) and \( P_2 \), we need to check the conditions for tangency for both parabolas.

Using the same method for \( P_2 \):
Substituting \( y^2 = -8x \) into the line equation gives:
\[ Lx + m(-8x) + n = 0 \]
This leads to a similar discriminant condition:
\[ 64m^2 - 32ln = 0 \]
From this, we can derive another relationship between \( l, m, n \).

### Step 5: Common Chord Condition
To find the common chord of the two parabolas, we set \( y = x^2 \) into \( y^2 = -8x \):
\[ (x^2)^2 = -8x \]
This results in:
\[ x^4 + 8x = 0 \]
Factoring gives:
\[ x(x^3 + 8) = 0 \]
The solutions are \( x = 0 \) and \( x = -2 \). The corresponding \( y \) values are \( y = 0 \) and \( y = 4 \).

The line joining these points \( (0, 0) \) and \( (-2, 4) \) can be found using the slope formula:
\[ \text{slope} = \frac{4 - 0}{-2 - 0} = -2 \]
Thus, the equation of the line is:
\[ y - 0 = -2(x - 0) \]
or equivalently:
\[ 2x + y = 0 \]

### Conclusion
The conditions derived from the discriminants and the analysis of rational points lead us to conclude that:
- Option (a) is correct: If \( l, m, n \) are odd integers, the line cannot intersect \( P_1 \) at a rational point.
- Option (b) is correct: The condition \( l^2 = 4mn \) holds for tangents.
- Option (c) is correct: If \( L \) is a common tangent to \( P_1 \) and \( P_2 \), then \( L + m + n = 0 \).
- Option (d) is correct: If \( L \) is the common chord of \( P_1 \) and \( P_2 \), then \( L - 2m + n = 0 \).