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Geometry (Circles) — Questions with Solutions

3 solved Geometry (Circles) questions from Quantitative Aptitude, as asked in SSC CGL and other SSC exams. Read the answer and explanation below, then practise the full set in the practice section.

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Q1. X, Y, and Z are three points on a plane with $XY=11\text{ cm}$, $YZ=13\text{ cm}$, and $XZ=24\text{ cm}$. The number of circles that travel through points X, Y, and Z is:

एक समतल पर X, Y, और Z तीन बिंदु इस प्रकार हैं कि $XY=11\text{ cm}$, $YZ=13\text{ cm}$, और $XZ=24\text{ cm}$ है। X, Y, और Z बिंदुओं से होकर गुजरने वाले वृत्तों की संख्या कितनी है?

  1. A. $0$
  2. B. $2$
  3. C. $1$
  4. D. $3$

Solution

Since $XY + YZ = XZ$ ($11 + 13 = 24$), the points are collinear. A circle cannot pass through three collinear points.

Q2. Let O be the centre of the circle and AB and CD are two parallel chords on the same side of the radius. OP is perpendicular to AB and OQ is perpendicular to CD. If AB = 10 cm, CD = 24 cm and PQ = 7 cm, then the diameter (in cm) of the circle is equal to:

  1. A. 26
  2. B. 13
  3. C. 12
  4. D. 24

Solution

Using Pythagorean theorem: R² = 5² + x², and R² = 12² + (x-7)². Solving gives x=12, R=13. Diameter = 2R = 26 cm.

Q3. Let C be a circle with centre O and P be an external point to C. Let PA and PB be two tangents to C with A and B being the points of tangency, respectively. If PA and PB are inclined to each other at an angle of 60°, then find ∠POA.

  1. A. 40°
  2. B. 60°
  3. C. 80°
  4. D. 30°

Solution

Tangents drawn from an external point are equally inclined to the line joining the center. Hence, ∠OPA = 60° / 2 = 30°. In right ΔOAP, ∠OAP = 90°, so ∠POA = 180° - 90° - 30° = 60°. Wait, looking at the answer keys, the official answer is 60°.