P and Q are 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 _________.

Correct — B. The two direct common tangents meet on the line of centres, at the point known as the external centre of similitude.

Drop the radii to the two points of contact on one tangent. Both are perpendicular to that tangent, so triangles APradius and AQradius are similar, sharing the angle at A.

Similarity gives AP ⁄ AQ = 7 ⁄ 3, the ratio of the radii.

A lies outside the segment PQ, beyond the smaller circle's centre Q, so the division is external, not internal.

A therefore divides PQ externally in the ratio 7 : 3 → option (b)

Why the others are wrong

• (a)
  externally in the ratio 3 : 7 — The ratio must be read in the order the points are named — AP to AQ, and P carries the 7 cm radius. Writing 3 : 7 describes a point beyond P, on the far side of the larger circle.
• (c)
  internally in the ratio 7 : 3 — Internal division puts the meeting point between P and Q, which is where the transverse common tangents cross. The direct tangents never cut the segment joining the centres.
• (d)
  internally in the ratio 3 : 7 — Wrong on both counts: internal belongs to the transverse tangents, and the ratio, read from P to Q, is 7 to 3 rather than 3 to 7.

Two circles have two centres of similitude, both sitting on the line of centres. The direct (external) common tangents concur at the external one, the transverse tangents at the internal one.

Each divides PQ in the ratio of the radii, r₁ : r₂. The difference is only where the point falls — outside the segment for the direct tangents, inside it for the transverse.

With radii 7 cm and 3 cm the ratio is 7 : 3 read from P's circle to Q's, and the direct tangents put A beyond Q, the smaller circle.

Key facts

• Direct common tangents meet at the external centre of similitude, on the line of centres extended.
• That point divides the line of centres externally in the ratio of the radii.
• Transverse common tangents meet at the internal centre of similitude, dividing the line of centres internally in the same ratio.
• With radii 7 cm and 3 cm the ratio is 7 : 3 whatever the distance between the centres.

Common traps

• Reversing the ratio to 3 : 7, which locates a point beyond P instead of beyond Q.
• Calling the division internal, which is the transverse tangents' answer.
• Searching the stem for a centre-to-centre distance that the ratio does not require.

No directly related past PYQ was found.