Mathematics
A and B are centres of circles of radii 9 cm and 2 cm such that AB = 17 cm and C is the centre of the circle of radius r cm which touches the above circles externally. If ∠ACB = 90°, write an equation in r and solve it.
Circles
3 Likes
Answer
From figure,
In △ABC,
By pythagoras theorem,
⇒ AB2 = AC2 + BC2
⇒ 172 = (r + 9)2 + (r + 2)2
⇒ 289 = r2 + 81 + 18r + r2 + 4 + 4r
⇒ 289 = 2r2 + 85 + 22r
⇒ 2r2 + 22r + 85 - 289 = 0
⇒ 2r2 + 22r - 204 = 0
⇒ 2(r2 + 11r - 102) = 0
⇒ r2 + 11r - 102 = 0
⇒ r2 + 17r - 6r - 102 = 0
⇒ r(r + 17) - 6(r + 17) = 0
⇒ (r - 6)(r + 17) = 0
⇒ r - 6 = 0 or r + 17 = 0
⇒ r = 6 or r = -17.
Since, radius cannot be negative.
⇒ r = 6 cm.
Hence, equation is r2 + 11r - 102 = 0 and r = 6 cm.
Answered By
2 Likes
Related Questions
A point P is 17 cm away from the centre of the circle and the length of the tangent drawn from P to the circle is 15 cm. Find the radius of the circle.
There are two concentric circles, each with centre O and of radii 10 cm and 26 cm respectively. Find the length of the chord AB of the outer circle which touches the inner circle at P.
Two circles touch each other externally at a point C and P is a point on the common tangent at C. If PA and PB are tangents to the two circles, prove that PA = PB.
Two circles touch each other internally. Prove that the tangents drawn to the two circles from any point on the common tangent are equal in length.