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*Result \(6\)*:

The two direct common tangents drawn to the circles are equal in length.

**Explanation**:

The two direct common tangents \(AC\) and \(BD\) from \(P\) drawn to the circles are equal in length.

\(\Rightarrow\) \(AC\) \(=\) \(BD\)

*Proof of the result*:

By the

**result**\(3\), we have:The lengths of the two tangents drawn from an exterior point to a circle are equal.

\(PA\) \(=\) \(PB\) and \(PC\) \(=\) \(PD\).

Subtract the above two equations.

\(PA\) \(-\) \(PC\) \(=\) \(PB\) \(-\) \(PD\)

\(AC\) \(=\) \(BD\)

Example:

In the above given figure if \(PB\) \(=\) \(9\) \(cm\) and \(AC\) \(=\) \(6\) \(cm\), find the length of the tangent \(PD\).

**Solution**:

By the result, we know that \(AC\) \(=\) \(BD\).

So, \(BD\) \(=\) \(6\) \(cm\).

From the figure it is observed that, \(PD\) \(=\) \(PB\) \(-\) \(BD\).

\(PD\) \(=\) \(9\) \(cm\) \(-\) \(6\) \(cm\)

\(PD\) \(=\) \(3\) \(cm\)

Therefore, the length of the tangent \(PD\) \(=\) \(3\) \(cm\).