In Fig. 9.20. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB
Solution:
Given, O is the centre of a circle of radius 5 cm.
T is a point outside the circle such that OT = 13 cm.
OT intersects the circle at E.
We have to find the length of AB if AB is the tangent to the circle at E
We know that tangents to a circle through an external point are equal.
From point A, the tangents to the circle are AP and AE
So, AP = AE
Let AP = AE be x.
We know that the radius of a circle is perpendicular to the tangent at the point of contact.
So, OP ⟂ PT and OQ ⟂ QT
Also, ∠OPT = ∠OQT = 90°
Considering triangle OPT,
∠P = 90°
So, OPT is a right triangle with P at right angle.
OT² = OP² + PT²
(13)² = (5)² + PT²
169 = 25 + PT²
PT² = 169 - 25
PT² = 144
Taking square root,
PT = 12 cm
Considering triangle AET,
∠E = 90°
So, AET is a right triangle with E at right angle.
By pythagoras theorem,
AT² = AE² + ET²
From the figure,
PT = PA + AT
12 = x + AT
AT = 12 - x
(12 - x)² = x² + ET²
ET² = (12 - x)² - x²
By using algebraic identity,
(a - b)² = a² - 2ab + b²
So, ET² = 144 - 24x + x² - x²
ET² = 144 - 24x
From the figure,
OT = OE + ET
13 = 5 + ET
ET = 13 - 5
ET = 8 cm
So, (8)² = 144 - 24x
64 = 144 - 24x
24x = 144 - 64
24x = 80
x = 80/24
x = 20/6
x = 10/3 cm
We know AB = 2(AE)
AB = 2(10/3)
AB = 20/3
AB = 6.6 cm
Therefore, the length of AB is 6.6 cm
✦ Try This: The radii of two concentric circles are 13cm and 8cm, AB is a diameter of bigger circle. BD is a tangent to the smaller circle touching it at D. Find the length AD.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 11
In Fig. 9.20. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, find the length of AB
Summary:
In Fig. 9.20. O is the centre of a circle of radius 5 cm, T is a point such that OT = 13 cm and OT intersects the circle at E. If AB is the tangent to the circle at E, the length of AB is 6.6 cm
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