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A tangent is drawn from a point p outside the circle, which touches the circle at point A and B, another tangent is drawn from a point E on the circumference on the circle, which touches the line PA and PB at C and D, if PC measures 8 cm, PD measures 9 cm and CD measures 7 cm, then find the radius of circle.

A tangent is drawn from a point p outside the circle, which touches the circle at point A and B, another tangent is drawn from a point E on the circumference on the circle, which touches the line PA and PB at C and D, if PC measures 8 cm, PD measures 9 cm and CD measures 7 cm, then find the radius of circle. Correct Answer 12/√5 cm

Let DB be ‘x’ cm and EC be ‘y’ cm

⇒ DB = ED and EC = CA

If ED = 3 cm and EC = 4 cm

⇒ PA = PB = 12 cm

Join PO and AO

⇒ AO = radius of circle and ∠PAO = 90°

In ΔPCD,

Let ∠CPD be 2α

⇒ cos2α = (82 + 92 – 72) /2 × 8 × 9

⇒ cos2α = 2/3

⇒ 2cos2α – 1 = 2/3

⇒ cosα = √5/√6

⇒ tanα = 1/√5

In ΔAPO,

⇒ tanα = 1/√5 = AO/12

∴ Radius of circle, AO = 12/√5

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