Question 5
Two circles with centers 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.

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Solution

Here, two circles are of radii OP = 3 cm and PO’ = 4 cm
These two circles intersect at P and Q.

Here, OP and PO’ are two tangents drawn at point P.
$∠$ OPO’ = $90^{\circ}$
[ tangents at any point of circle is perpendicular to radius through the point of contact]
Join OO’ and PN
In right angled $△$ OPO’
$(O O^{'})^{2} = (O P)^{2} + (P O^{'})^{2}$ [ by Pythagoras theorem]
i.e $(H y p o t e n u s e)^{2}$ = $(B a s e)^{2}$ + $(P e r p e n d i c u l a r)^{2}$
$= (3)^{2} + (4)^{2} = 25$
$\Rightarrow$ OO’ = 5cm
Also. PN $⊥$ OO’
Let ON = x, then NO’ = 5 - x
In right angled $∠$ OPN ,
$(O P)^{2} = (O N)^{2} + (N P)^{2}$ [ by Pythagoras theorem]
$\Rightarrow (N P)^{2} = 3^{2} - x^{2} = 9 - x^{2}$
And in right angled $△$ PNO'
$(P O^{'})^{2} = (P N)^{2} + (N O^{'})^{2}$ [by Pythagoras theorem]
$\Rightarrow (4)^{2} = (P N)^{2} + (5 - x)^{2}$
$\Rightarrow (P N)^{2} = 16 - (5 - x)^{2}$
From Eqs. (i) and (ii)
$9 - x^{2} = 16 - (5 - x)^{2} \Rightarrow 7 + x^{2} - (25 + x^{2} - 10 x) = 0 \Rightarrow 10 x = 18 ∴ x = 1.8$
Again in right angled $△$ OPN
$O P^{2} = (O N)^{2} + (N P)^{2}$ [ by Pythagoras theorem]
$\Rightarrow 3^{2} = (1.8)^{2} + (N P)^{2}$
$\Rightarrow (N P)^{2} = 9 - 3.24 = 5.76$
$∴$ (NP) = 2.4
$∴$ Length of common chord.
PQ = 2 PN = 2 $\times$ 2.4 = 4.8 cm.

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Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.

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Question 5 Two circles with centers 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.

Question 5
Two circles with centers 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.