Segments of Secants, Chord and Tangent
Segments of Secants and Chord: Secants AB and CD intersect inside the circle in fig (a) and outside the circle in fig (b).
From the figure, we have
∠DCB = ∠DAB
and ∠ADC = ∠ABC
Hence ΔADE ~ ΔCBE
∴ \(\frac{AE}{CE}=\frac{DE}{BE}\).
AE x BE = CE x DE
Segments of Tangent: In fig (c), AD is secant and AB is tangent. Therefore,
ΔABD ~ ΔACB
∴ \(\frac{AB}{AC}=\frac{AD}{AB}\).
AB² = AC x AD
Example: If a line is drawn through a fixed-point P (α, β) to cut the circle x² + y² = a² at A and B, then find the value of PA.PB.
From the figure,
x² + y² = a²
PA. PB = constant
Also PA. PB = PC²
But PC² = OP² – OC²
= α² + β² – a²
PA. PB = α² + β² – a².