Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
Let $ABC$ be a triangle. Let $A$ be the point $(1, 2)$, $y = x$ is the perpendicular bisector of $AB$ and $x - 2y + 1 = 0$ is the angle bisector of angle $C$. If the equation of $BC$ is given by $ax + by - 5 = 0$ then the value of $a + b$ is ______.
Step-by-Step Solution
Key Concept: Reflection of a point across a line is found by extending the perpendicular through the point to an equal distance on the opposite side.
To find the reflection of point $A$ across the line $x - 2y + 1 = 0$, we use the parametric form with parameter $t$. The perpendicular from $A$ to the line gives $t = 2$, yielding the foot of perpendicular. Reflecting across this foot using the formula $A' = 2 \times \text{(foot)} - A$ gives $A' = \left(\frac{9}{5}, \frac{2}{5}\right)$. The line $BC$ through points $B(2,1)$ and $C(1,2)$ has equation $3x - y - 5 = 0$, verified by $a + b = 3 - 1 = 2$.
Correct Answer: 2