Problems on Common Tangents to Two Circles
Here we will solve different types of problems on common tangents to two circles.
1. There are two circles touch each other externally. Radius of the first circle with centre O is 8 cm. Radius of the second circle with centre A is 4 cm. Find the length of their common tangent BC.
Solution:
Join O to A and B. Join A to C. Draw DA ⊥ OB.
Now DA = BC, as they are opposite sides of rectangle ACBD.
OA = 8 cm + 4 cm
= 12 cm.
OD = 8 cm – 4cm
= 4 cm.
Therefore, DA = \(\sqrt{OA^{2} - OD^{2}}\)
= \(\sqrt{12^{2} - 4^{2}}\) cm
= \(\sqrt{144 - 16}\) cm
= \(\sqrt{128}\) cm
= 8√2 cm
Therefore, BC = 8√2 cm.
2. Prove that a transverse common tangent drawn to two circles divides the line joining their centres into the ratio of their radii.
Solution:
Given: Two circles with centres O and P, and radii OX and PY respectively. The transverse common tangent XY touches them at X and Y respectively. XY cuts OP at T.
To prove: \(\frac{OT}{TP}\) = \(\frac{OX}{PY}\).
Proof:
|
Statement |
Reason |
|
1. In ∆XOT and ∆YPT, (i) ∠OXT = ∠PYT = 90 ° (ii) ∠OTX = ∠PTY. |
1. (i) Tangent ⊥ Radius. (ii) Vertically opposite angles. |
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2. ∆XOT ∼ ∆YPT |
2. By A – A criterion of similarity. |
|
3. Therefore, \(\frac{OT}{TP}\) = \(\frac{OX}{PY}\). (Proved) |
3. Corresponding sides of similar triangles are proportional. |
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