Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ
Solution:
Given, two circles with centres O and O' have radii 3 cm and 4 cm.
Two circles intersect at two points P and Q.
OP and O’P are the tangents to the two circles.
We have to find the length of the common chord PQ.
We know that the radius of a circle is perpendicular to the tangent at the point of contact.
So, ∠OPO’ = 90°
Considering triangle OPO’,
OPO’ is a right triangle with P at right angle.
(OO’)² = (OP)² + (O’P)²
From the figure,
OP = radius of circle = 3 cm
O’P = radius of other circle = 4 cm
(OO’)² = (3)² + (4)²
(OO’)² = 9 + 16
(OO’)² = 25
Taking square root,
OO’ = 5 cm
Let ON = x cm
So, O’N = 5 - x cm
In triangle ONP,
By pythagoras theorem,
(OP)² = (ON)² + (PN)²
(3)² = (x)² + (PN)²
9 = x² + (PN)²
PN² = 9 - x² -------------------------- (1)
In triangle O’NP,
(O’P)² = (O’N)² + (PN)²
(4)² = (5 - x)² + PN²
PN² = 16 - (5 - x)²
By using algebraic identity,
(a - b)² = a² - 2ab + b²
PN² = 16 - (25 -10x + x²)
PN² = 16 - 25 + 10x - a²
PN² = -x² + 10x - 9 ----------------- (2)
Comparing (1) and (2),
9 - x² = -x² + 10x - 9
9 = 10x - 9
10x = 9 + 9
10x = 18
x = 18/10
x = 1.8
Substitute the value of x in (1),
PN² = 9 - (1.8)²
PN² = 9 - 3.24
PN² = 5.76
Taking square root,
PN = 2.4 cm
We know, PQ = 2PN
PQ = 2(2.4)
PQ = 4.8 cm
Therefore, the length of the chord PQ is 4.8 cm
✦ Try This: Two circles with centers O and O’ intersect at two points A and B. A line PQ is drawn parallel to OO’ through A(or B) intersecting the circles at P and Q. Prove that PQ = 2 OO’.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 5
Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ
Summary:
Two circles with centres O and O' of radii 3 cm and 4 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. The length of the common chord PQ is 4.8 cm
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