Q: X, Y, and Z are three points on a plane with $XY = 11$ cm, $YZ = 13$ cm, and $XZ = 24$ cm. The number of circles passing through points X, Y, and Z is:

0. Here $XY + YZ = 11 + 13 = 24 = XZ$, so the three points are collinear. A circle cannot pass through three distinct collinear points, hence the number of circles is zero.

Question 1

P and Q are the centres of two circles whose radii are 7 cm and 3 cm, respectively. If the direct common tangents to the circles meet PQ extended at A, then A divides PQ:

Accepted Answer

externally in the ratio 7: 3. When the direct common tangents meet the line joining the centres externally, the point divides the segment externally in the ratio of the radii. Since the radii are 7 cm and 3 cm, A divides PQ externally in the ratio 7:3.

Question 2

The radii of two circles are 5 cm and 10 cm, and the distance between their centres is 17 cm. Find the length of the transverse common tangent.

Accepted Answer

8 cm. For two circles, the length of the transverse common tangent is $\sqrt{d^2-(r_1+r_2)^2}$. Substituting $d=17$, $r_1=5$, and $r_2=10$, we get $\sqrt{17^2-15^2} = \sqrt{289-225} = \sqrt{64} = 8$. Hence, the answer is 8 cm.

Question 3

Two circles of radii 18 cm and 12 cm touch each other externally. Find the length (in cm) of their direct common tangent.

Accepted Answer

12√6. Since the circles touch externally, the distance between their centers is 18 + 12 = 30 cm. For a direct common tangent, length = $\sqrt{30^2-(18-12)^2}=\sqrt{900-36}=\sqrt{864}=12\sqrt{6}$.

Question 4

X, Y, and Z are three points on a plane with $XY = 11$ cm, $YZ = 13$ cm, and $XZ = 24$ cm. The number of circles passing through points X, Y, and Z is:

Accepted Answer

0. Here $XY + YZ = 11 + 13 = 24 = XZ$, so the three points are collinear. A circle cannot pass through three distinct collinear points, hence the number of circles is zero.

SSC CGL (Prelims) Maths: Circles questions with solutions

Questions