Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
In a triangle $ABC$, the coordinates of $A$ is $(1, 2)$ and the equations to the medians through $B$ and $C$ are $x + y = 5$ and $x = 4$. If coordinate of $B$ is $(x_1, y_1)$ and co-ordinate of $C$ is $(x_2, y_2)$ then $x_1y_2 + x_2y_1$ equals____.
Step-by-Step Solution
Key Concept: Midpoints of segments from a vertex must lie on specified lines, which constrains the coordinates of the other vertices.
Point $B$ on line $x + y = 5$ has form $(h, 5-h)$ and point $C$ on line $x = 4$ has form $(4, k)$. Using the midpoint of $AB$ lies on $x = 4$ gives $h = 7$, so $B = (7, -2)$. Using the midpoint of $AC$ lies on $x + y = 5$ gives $k = 3$, so $C = (4, 3)$.
Correct Answer: 13