Question

The locus of the point which forms a right-angled triangle with the fixed points \((2,3)\) and \((5,1)\) is

• A. A circle or a pair of parallel lines
• B. A pair of parallel lines which are parallel to the line joining the given points
• C. A circle having the line joining the given points as a chord
• D. The perpendicular bisector of the line joining the given points

Hint:

Whenever a line segment subtends a right angle at a moving point, immediately think of Thales' theorem: the locus is a circle having that segment as diameter.

Answer 1

The Correct Option is A

Step 1: Understand the question.
We have two fixed points and a third moving point. The three points together make a right-angled triangle. We have to describe the path (locus) traced by that moving point.

Step 2: Name the points.
Let the fixed points be $A(2,3)$ and $B(5,1)$. Let the moving point be $P(x,y)$. The triangle is $APB$, and one of its angles is a right angle.

Step 3: Case where the right angle is at $P$.
A well-known result (the angle in a semicircle) says: if a fixed segment $AB$ subtends a right angle at $P$, then $P$ moves on a circle that has $AB$ as its diameter. So this case gives a circle.

Step 4: Case where the right angle is at $A$ or at $B$.
If the right angle is at $A$, then $PA$ is perpendicular to $AB$, so $P$ lies on the line through $A$ at right angles to $AB$. If the right angle is at $B$, then $P$ lies on the line through $B$ at right angles to $AB$. These two lines have the same slope, so they are a pair of parallel lines.

Step 5: Put the cases together.
Collecting every possibility, the moving point lies either on the circle (right angle at $P$) or on one of the two parallel lines (right angle at $A$ or $B$). So the full locus is a circle together with a pair of parallel lines.

Step 6: Match with the options.
The complete description is "a circle or a pair of parallel lines," which is option 1.
\[ \boxed{\text{A circle or a pair of parallel lines}} \]