Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade None
Question:
Two sides of a triangle have the joint equation $(x - 3y + 2)(x + y - 2) = 0$, the third side which is variable always passes through the point $(-5, -1)$, then the possible values of slope of third side such that origin is an interior point of triangle is/are:
$\frac{-4}{3}$
$\frac{-2}{3}$
$\frac{-1}{3}$
$\frac{1}{6}$
Step-by-Step Solution
Key Concept: A point lies inside a triangle if and only if it satisfies the correct inequalities with respect to all three bounding lines.
For the origin to lie inside triangle with vertices formed by three lines $x + y = 2$, $x - 3y + 2 = 0$, and $m = -1$ (or $m = \frac{1}{5}$), the origin must satisfy all three inequalities consistently. Testing the origin $(0,0)$ against the inequality constraints from the three lines yields $m \in (-1, \frac{1}{5})$.
Correct Answer: 2,3,4