Question

A line is drawn through a fix point P(`alpha, beta`) to cut the circle `x^2 + y^2 = r^2` at A and B. Then PA.PB is equal to :

Comments

Why is PA.PB = PC²

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It is a theorem that if a line segment from a fixed outer point to a circle intersect the circle at two point then the product of distances of the outer point from these point is equal to the square of radius.

Answer 1

From the figure,
PA.PB`=` Constant
Also, `PA.PB=PC^(2)`
But `PC^(2)=OP^(2)-OC^(2)`
`=alpha^(2)+beta^(2)-a^(2)`
Hence, `PA.PB=alpha^(2)+beta^(2)-a^(2)`