Question

In the given figure two circles touch each other at the point C. Prove that the common tangent at P and Q.

Answer 1

Here we are given that two circles touch each other at point C

Since, we know that lengths of tangent drawn from an external point to a circle are equal.

So consider circle with centre A

According to external point theorem

RP = RC

Similarly consider circle with centre B

RQ = RC

∴ RP = RC and RC = RQ

⇒ RP = RQ

Hence we can say that R is the mid-point of P and Q

Hence the common tangent to the circles at C, bisects the common tangent at P and Q.

Answer 2

In the given figure, PR and CR are both tangents drawn to the same circle from an external point R
`:." "PR=CR" "…(1)`
Also, QR and CR are both tangent drawn to the same circle from an external point R.
`:." "QR=CR`
From (1) and (2) we get
`PR=QR`
`:.` R is the mid-point of PQ
i.e., the common tangent at C bisects the common tangents at P and Q.
Hence Proved.