# 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

**☛ Related Questions:**

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