The points `(a t_1^2, 2 a t_1),(a t_2^2, 2a t_2) and (a ,0)` will be collinear, if

Options

A. 2
B. `-1`
C. 1
D. None of these

Answer

To determine the conditions under which the points \((a t_1^2, 2 a t_1)\), \((a t_2^2, 2 a t_2)\), and \((a, 0)\) are collinear, we can use the concept of the area of the triangle formed by these three points. If the area is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Identify the Points:**
Let the points be:
- \( A = (a t_1^2, 2 a t_1) \)
- \( B = (a t_2^2, 2 a t_2) \)
- \( C = (a, 0) \)

2. **Set Up the Determinant:**
The points \( A \), \( B \), and \( C \) will be collinear if the following determinant is zero:
\[ \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0 \]
Substituting the coordinates:
\[ \begin{vmatrix} a t_1^2 & 2 a t_1 & 1 \\ a t_2^2 & 2 a t_2 & 1 \\ a & 0 & 1 \end{vmatrix} = 0 \]

3. **Calculate the Determinant:**
Expanding the determinant:
\[ = a t_1^2 \begin{vmatrix} 2 a t_2 & 1 \\ 0 & 1 \end{vmatrix} - 2 a t_1 \begin{vmatrix} a t_2^2 & 1 \\ a & 1 \end{vmatrix} + 1 \begin{vmatrix} a t_2^2 & 2 a t_2 \\ a & 0 \end{vmatrix} \]
Calculating each of the 2x2 determinants:
- First determinant: \( 2 a t_2 \cdot 1 - 0 \cdot 1 = 2 a t_2 \)
- Second determinant: \( a t_2^2 \cdot 1 - a \cdot 1 = a(t_2^2 - 1) \)
- Third determinant: \( a t_2^2 \cdot 0 - 2 a t_2 \cdot a = -2 a^2 t_2 \)

Substituting back:
\[ = a t_1^2 (2 a t_2) - 2 a t_1 (a(t_2^2 - 1)) - 2 a^2 t_2 \]
Simplifying:
\[ = 2 a^2 t_1^2 t_2 - 2 a^2 t_1 (t_2^2 - 1) - 2 a^2 t_2 \]
\[ = 2 a^2 (t_1^2 t_2 - t_1 t_2^2 + t_1 - t_2) \]

4. **Set the Determinant to Zero:**
For collinearity, we set the expression to zero:
\[ 2 a^2 (t_1^2 t_2 - t_1 t_2^2 + t_1 - t_2) = 0 \]
Since \(2 a^2\) cannot be zero (assuming \(a \neq 0\)), we can simplify to:
\[ t_1^2 t_2 - t_1 t_2^2 + t_1 - t_2 = 0 \]

5. **Factor the Expression:**
Rearranging gives:
\[ (t_1 - t_2)(t_1 + t_2) = 0 \]
This implies:
- \(t_1 = t_2\) or
- \(t_1 + t_2 = 0\) (which means \(t_1 t_2 = -1\))

### Final Conditions:
Thus, the points will be collinear if:
1. \(t_1 = t_2\)
2. \(t_1 t_2 = -1\)