Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
A line $L_1$ intersect $x$ and $y$ axes at $P$ and $Q$ respectively. Another line $L_2$, perpendicular to $L_1$, cuts $x$ and $y$ axes at $T$ and $S$ respectively. The locus of the point of intersection of the lines $PS$ and $QT$ is a circle passing through the
origin
point P
point Q
point T
Step-by-Step Solution
Key Concept: The locus of the orthocenter's foot of altitude in a triangle with two fixed vertices and fixed right angles is a circle with those vertices as diameter.
Since $T$ is the orthocenter of triangle $OPS$ and $\angle ORP = \angle OOP = 90°$, the points $Q$, $R$, $P$ are concyclic with $PQ$ as diameter. Since $O$ is also the orthocenter and lies on this circle, the locus of $R$ forms a circle having $PQ$ as diameter.
Correct Answer: 1,2,3