JAMB 1999 · UME · Q28

In the figure, $TZ$ is a tangent to the circle $QPZ$ and $TPQ$ is a straight line. Find $x = TP$ if $TZ = 6$ units and $PQ = 9$ units.

Not to scale.

Worked solution (try it first)

Tangent–secant rule: the tangent squared equals (outside part) × (whole secant).
So $TZ^2 = TP \times TQ$ , with $TQ = x + 9$ .
So $36 = x(x + 9)$ , which rearranges to $x^2 + 9x - 36 = 0$ .
Factorise: $(x + 12)(x - 3) = 0$ .
A length is positive, so $x = 3$ , option A.