Question : A secant PAB is drawn from an external point P to the circle with centre O, intersecting it at A and B. If OP = 17 cm, PA = 12 cm, and PB = 22.5 cm, then the radius of the circle is:
Option 1: $2 \sqrt{3} {~cm}$
Option 2: $\sqrt{19} {~cm}$
Option 3: $\sqrt{17} {~cm}$
Option 4: $3 \sqrt{2} {~cm}$
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Correct Answer: $\sqrt{19} {~cm}$
Solution :
Given:
OP = 17 cm, PA=12 cm and PB = 22.5 cm
We know,
Two chords AB and CD when extended meet at point P, then
PA × PB = PC × PD
Let the radius be x.
⇒ PC = 17 – x and PD = 17 + x
According to Question-
PA × PB = PC × PD
⇒ 12 × 22.5 = (17 – x)(17 + x)
⇒ 270 = 289 – x$^2$
⇒ x$^2$ = 19
⇒ x = $\sqrt{19}$ cm
Hence, the correct answer is $\sqrt{19}$ cm.
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