Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
In a $\triangle ABC$, the vertex $A$ is $(1, 1)$ and orthocenter is $(2, 4)$. If the sides $AB$ and $BC$ are members of the family of straight lines $ax + by + c = 0$. Where $a, b, c$ are in A.P., then the coordinates of vertex $C$ are $(h, k)$. The value of $2h + 9k$ is ________.
Step-by-Step Solution
Key Concept: A family of lines satisfying a constraint passes through a fixed point, which can be found by appropriate substitution.
Given $a + c - 2b = 0$, the line $ax + by + c = 0$ passes through fixed point $(1, -2)$. This gives $B \equiv (1, -2)$. The slope of $BC$ is $\frac{2-(-1)}{4-1} = 1$. Using the condition on slopes and solving the resulting equations, we find $\lambda = -17$, giving $C \equiv (-17, 4)$. The centroid equation yields $2h + 9k = 2$.
Correct Answer: 2