There are two concentric circles with centre P. The larger circle has radius PQ = 10 cm, and the smaller circle has radius PR = 8 cm. TQ is a tangent to the larger circle from an external point T (touching it at Q), and TR is a tangent to the smaller circle from the same point T (touching it at R). If TQ = 24 cm, find the value of (TR – PR)/PQ.
\(\frac{3\sqrt{17} - 4}{5}\)
A tangent is perpendicular to the radius at the point of contact, so triangles TQP and TRP are right-angled at Q and R.
In triangle TQP: \(TP^{2} = TQ^{2} + PQ^{2} = 24^{2} + 10^{2} = 576 + 100 = 676\), so TP = 26 cm.
In triangle TRP: \(TR^{2} = TP^{2} - PR^{2} = 676 - 64 = 612\), so \(TR = \sqrt{612} = 6\sqrt{17}\) cm.
Then \(\frac{TR - PR}{PQ} = \frac{6\sqrt{17} - 8}{10} = \frac{3\sqrt{17} - 4}{5}\).
Check: \((6\sqrt{17})^{2} + 8^{2} = 612 + 64 = 676 = 26^{2}\), so TR and PR fit the same TP = 26 cm found from the larger circle.
Hence, \(\frac{3\sqrt{17} - 4}{5}\).
Similar Questions
Suppose C and D are centres of two circles of radii 16 cm and 9 cm respectively, and distance between the centres is 25 cm. Find the length of direct common tangent.
Two tangents AB and BC are drawn to the circle with centre O, such that ∠ABC = 60°. If AB = 8.7 cm, find the length of OB.
Which of the following is NOT correct regarding the properties of a circle?
From an external point P, two tangents PA and PB are drawn to a circle with center O. A secant passing through P cuts the circle at points C and D such that PC = 4 cm and PD = 8 cm. If the radius of the circle is 6 cm, find the length of the tangent PA.
A, B, C and D are four points in order on a circle such that C is exactly opposite to A and D is exactly opposite to B. Which of the followings is NOT true?
Important Questions from Circles, Chords and Tangents
AB is a chord of a circle with centre O and P is any point on the circle. If ∠APB = 112°, then what is the measure of ∠OAB ?
The distance between the centres of two circles is 24 cm. If the radius of the two circles are 4 cm and 8 cm, then what is the sum of the lengths (in cm) of the direct common tangent and the transverse common tangent?
An equilateral triangle ABC and a scalene triangle DBC are inscribed in a circle on same side of the arc. what is ∠BDC equal to?
What can be the maximum number of common tangent which can be drawn to two non-intersecting circles?
There are two identical circles of radius 10 cm each. If the length of the direct common tangent is 26 cm, then what is the length (in cm) of the transverse common tangent?
