First slide

Definition of a circle

Question

A line is drawn through a fixed point $P (α, β)$ to cut the circle
$x^{2} + y^{2} = r^{2}$ at $A$ and $B$ .Then $P A \cdot P B$ is equal to

Moderate

A

$(α + β)^{2} - r^{2}$
B

$α^{2} + β^{2} - r^{2}$
C

$(α - β)^{2} + r^{2}$
D

none of these

Solution

The equation of any line through $P (α, β)$ is
$\frac{x - α}{\cos θ} = \frac{y - β}{\sin θ} = k$ (say)
Any point on this line is $(α + k \cos θ, β + k \sin θ) .$ This point lies on the given circle if
$(α + k \cos θ)^{2} + (β + k \sin θ)^{2} = r^{2}$
$\Rightarrow k^{2} + 2 k (α \cos θ + β \sin θ) + α^{2} + β^{2} - r^{2} = 0$ (i)
This equation, being quadratic in $k,$ gives two values of $k$ and hence the distances of two points A and B on the circle from the point P.
Let $P A = k_{1}, P B = k_{2},$ where $k_{1}, k_{2}$ are the roots of equation (i)
Then,
$PAPB = k_{1} k_{2} = α^{2} + β^{2} - r^{2}$
ALITER PAPB is the power of the point $P (α, β)$ ) with respect to the circle $x^{2} + y^{2} = r^{2}$ . Therefore ,
$P A P B = α^{2} + β^{2} - r^{2}$

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