Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
Let $x_1$ and $y_1$ be the roots of $x^2 + 8x - 2009 = 0$; $x_2$ and $y_2$ be the roots of $3x^2 + 24x - 2010 = 0$ and $x_3$ and $y_3$ be the roots of $9x^2 + 72x - 2011 = 0$. The points $A(x_1, y_1)$, $B(x_2, y_2)$ and $C(x_3, y_3)$:
can not lie on a circle
form a triangle of area 2 sq. units
form a right angled triangle
are collinear
Step-by-Step Solution
Key Concept: Three points are collinear if and only if the determinant formed by their coordinates (with a column of 1s) equals zero.
The determinant condition $D = \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = 0$ is applied using column operations. Substituting $C_1 \to C_1 + C_2$ transforms the determinant to $\begin{vmatrix} -8 & y_1 & 1 \\ -8 & y_2 & 1 \\ -8 & y_3 & 1 \end{vmatrix} = 0$, confirming collinearity when the determinant vanishes.
Correct Answer: 1,4