UPSC Prelims 2003 Q141: Science & Technology – Quantitative Aptitude
‘A’ and ‘B’ are two fixed points in a field. A cyclist C moves such that ACB is always a right
angle. In this context,
which one of the following statements is correct?
- (a) The path followed by the cyclist is an ellipse
- (b) The path followed by the cyclist is a circle
- (c) The path followed by the cyclist is an exponential curve
- (d) This type of motion is not possible
Answer: (b) The path followed by the cyclist is a circle
Explanation
Locus of a cyclist keeping a right angle at the fixed points
| Item | Result |
|---|---|
| Condition | Angle ACB is always 90 degrees for fixed points A and B |
| Thales' theorem | The angle in a semicircle is a right angle |
| Locus of C | A circle with AB as diameter |
By the converse of Thales' theorem, a point C that always sees AB at a right angle moves on a circle whose diameter is AB.
Answer: (b) The path followed by the cyclist is a circle.
Why each option is right or wrong
- (a) Incorrect. An ellipse is not the locus of a right angle.
- (b) CORRECT. The locus is a circle with AB as its diameter.
- (c) Incorrect. An exponential curve is not the locus.
- (d) Incorrect. The motion is possible along the circle.
Key point
Right angle subtended by a fixed segment: circle on that segment as diameter.
Remember
A point seeing a segment at a right angle lies on a circle with that segment as diameter.
Common mistakes
- Choosing an ellipse because there are two fixed points.
- Thinking that the motion is impossible.
- Forgetting Thales' theorem about the semicircle.
Exam tip
Angle in a semicircle is 90 degrees, and the converse.
Sources
- NCERT Class IX Mathematics — Circles
- NCERT Class XI Mathematics — Conic Sections