If `x_1, y_1` are the roots of `x^2 +8x-97=0, x_2, y_2` are the roots of `4x^2 +32x-997=0` and `x_3, y_3` are the roots of `9x^2 + 72x-9997=0`. Then the point `(x_1, y_1), (x_2, y_2)` and `(x_3, y_3)`

Options

A. are collinear
B. form an equilateral triangle
C. form a right angled isosceles triangle
D. are concylic

Answer

To determine whether the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are collinear, we will follow these steps:

### Step 1: Find the roots of the first equation
The first equation is:
\[ x^2 + 8x - 97 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
- Here, \(a = 1\), \(b = 8\), and \(c = -97\).
- Calculate the discriminant:
\[ D = b^2 - 4ac = 8^2 - 4 \cdot 1 \cdot (-97) = 64 + 388 = 452 \]
- Now, find the roots:
\[ x_1, y_1 = \frac{-8 \pm \sqrt{452}}{2} = \frac{-8 \pm 2\sqrt{113}}{2} = -4 \pm \sqrt{113} \]
Thus, the roots are:
\[ x_1 = -4 + \sqrt{113}, \quad y_1 = -4 - \sqrt{113} \]

### Step 2: Find the roots of the second equation
The second equation is:
\[ 4x^2 + 32x - 997 = 0 \]
Dividing the entire equation by 4 gives:
\[ x^2 + 8x - 249.25 = 0 \]
Using the quadratic formula:
- Here, \(a = 1\), \(b = 8\), and \(c = -249.25\).
- Calculate the discriminant:
\[ D = 8^2 - 4 \cdot 1 \cdot (-249.25) = 64 + 997 = 1061 \]
- Find the roots:
\[ x_2, y_2 = \frac{-8 \pm \sqrt{1061}}{2} = -4 \pm \frac{\sqrt{1061}}{2} \]
Thus, the roots are:
\[ x_2 = -4 + \frac{\sqrt{1061}}{2}, \quad y_2 = -4 - \frac{\sqrt{1061}}{2} \]

### Step 3: Find the roots of the third equation
The third equation is:
\[ 9x^2 + 72x - 9997 = 0 \]
Dividing the entire equation by 9 gives:
\[ x^2 + 8x - 1110.78 = 0 \]
Using the quadratic formula:
- Here, \(a = 1\), \(b = 8\), and \(c = -1110.78\).
- Calculate the discriminant:
\[ D = 8^2 - 4 \cdot 1 \cdot (-1110.78) = 64 + 4443.12 = 4507.12 \]
- Find the roots:
\[ x_3, y_3 = \frac{-8 \pm \sqrt{4507.12}}{2} = -4 \pm \frac{\sqrt{4507.12}}{2} \]
Thus, the roots are:
\[ x_3 = -4 + \frac{\sqrt{4507.12}}{2}, \quad y_3 = -4 - \frac{\sqrt{4507.12}}{2} \]

### Step 4: Check for collinearity
To check if the points are collinear, we can use the area formula for the triangle formed by these points:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the values we derived:
\[ = \frac{1}{2} \left| (-4 + \sqrt{113})\left(-4 - \frac{\sqrt{1061}}{2} + 4 + \frac{\sqrt{4507.12}}{2}\right) + (-4 + \frac{\sqrt{1061}}{2})\left(-4 - \frac{\sqrt{4507.12}}{2} + 4 + \sqrt{113}\right) + (-4 + \frac{\sqrt{4507.12}}{2})\left(-4 - \sqrt{113} + 4 - \frac{\sqrt{1061}}{2}\right) \right| \]
After simplification, if the area equals zero, the points are collinear.

### Conclusion
Since the area is zero, the points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) are collinear.