Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
In a triangle $ABC$, the bisector of angles $B$ and $C$ lies along the lines $y = x$ and $y = 0$. If $A$ is $(1,2)$ then $\sqrt{10} d(A, BC)$ equal (where $d(A, BC)$ denotes the perpendicular distance of $A$ from $BC$).
Step-by-Step Solution
Key Concept: The perpendicular distance from a point to a line uses the standard point-to-line distance formula $ rac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$.
The image of point $A(1,2)$ with respect to lines $y = x$ and $y = 0$ gives points $(2,1)$ and $(1,-2)$ respectively. The equation of line $BC$ is $y = 3x - 5$. The perpendicular distance from $A$ to $BC$ is calculated using the formula $d(A, BC) = rac{|3 - 2 - 5|}{\sqrt{10}} = rac{4}{\sqrt{10}}$. Therefore, $\sqrt{10} \cdot d(A, BC) = 4$.
Correct Answer: 4