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If the angle of intersection at a point...

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is `90^(@)`, then the length (in cm) of their common chord is

A

`(13)/(5)`

B

`(120)/(13)`

C

`(60)/(13)`

D

`(13)/(2)`

Text Solution

Verified by Experts

Let the length of common chord = AB = 2AM = 2x

Now, `C_(1)C_(2)=sqrt(AC_(1)^(2)+AC_(2)^(2))" "...(i)`
" " [`therefore` circles intersect each other at `90^(@)`]
and `C_(1)C_(2)=C_(1)M+MC_(2)`
`rArr C_(1)C_(2)=sqrt(12^(2)-AM^(2))+sqrt(5^(2)-AM^(2))" "...(ii)`
From Eqs. (I) and (ii) we get
`sqrt(AC_(1)^(2)+AC_(2)^(2))=sqrt(144-AM^(2))+sqrt(25-AM^(2))`
`rArr sqrt(144+25)=sqrt(144-x^(2))+sqrt(25-x^(2))`
`rArr 13 = sqrt(144-x^(2))+sqrt(25-x^(2))`
On squaring both sides, we get
`169=144-x^(2)+25-x^(2)+2sqrt(144-x^(2))sqrt(25-x^(2))`
`rArr 13 = sqrt(44-x^(2))sqrt(25-x^(2))`
Again on squaring both sides, we get
`x^(4)=(144-x^(2))=(144xx25)-(25+144)x^(2)+x^(4)`
`rArr x^(2)=(144xx25)/(169)rArr x=(12xx5)/(13)=(60)/(13)cm`
Now, lenghth of common chord `2x = (120)/(13)` cm
Alteranative Solution
Given, `AC_(1)=12 cm and AC_(2) = 5 cm`
In `DeltaC_(1)AC_(2)`,
`C_(1)C_(2)=sqrt((C_(1)A)^(2)+(AC_(2))^(2))" "[therefore angleC_(1)AC_(2)=90^(@), " because circles intersects each other at " 90^(@)]`
`=sqrt((12)^(2)+(5)^(2))=sqrt(144+25)`
`=sqrt169=13cm`
Now, area of `DeltaC_(1)AC_(2)=(1)/(2)AC_(1)xxAC_(2)`
`=(1)/(2)xx12xx5=30 cm^(2)`
Also, area of `DeltaC_(1)AC_(2)=(1)/(2)C_(1)C_(2)xxAM`
`=(1)/(2)xx13xx(AB)/(2)" "[therefore AM=(AB)/(2)]`
`therefore (1)/(4)xx13xxAM=30 cm`
`AM = (120)/(13) cm`
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