Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
A line intersects the x-axis at $A(7, 0)$ and y-axis $B(0, -5)$. A variable line $PQ$ perpendicular to $AB$ intersects the x-axis at $P$ and the y-axis at $Q$. If $AQ$ and $BP$ intersect at $R$, show that the locus of $R$ is $x^2 + y^2 - ax + by = 0$. Then the value of $(a - b)$ is ______.
Step-by-Step Solution
Key Concept: The orthocenter is the intersection of altitudes; when two sides lie on coordinate axes, the third altitude's foot determines a specific locus.
Since $P$ is the orthocenter of triangle $ABQ$ with $x$ and $y$ axes as two sides, point $R$ (foot of the third altitude) lies on $AB$. The orthocenter property gives $(\frac{y+5}{x})(\frac{y}{x-7}) = -1$, which simplifies to $x^2 + y^2 - 7x + 5y = 0$.
Correct Answer: 2