O and O are the centres of two circles

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.

By pythagoras theorem,

(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

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

☛ Related Questions:

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In the given figure, O and O' are centres of two circles intersecting at B and C. ACD is a straight line, find x.

It is given that

Two circles having center O and O' and ∠AOB = 130°

And AC is diameter of circle having center O

We have

`angle ACB =1/2 angleAOB = 65°`

`angleDCB = 180° - angleACB`

= 180° - 65°

= 115°

Now, reflex `angleBO'D = 2 angleBCD`

`360° - x° = 2 xx 115 `

= 230°