The measure of the chord intercepted by circle `x^2 + y^2 = 9` and the line x-y+2= 0 is

Options

A. `sqrt(28)`
B. `2sqrt(5)`
C. 7
D. 5

Answer

A

Explanation

Given, equation of circle is
`x^(2)+y^(2)=9`

Here, centre `-=(0,0)` and radius = 3 units
As, chord is intercepted by circle and line `x-y+2=0`.
Draw `" ""OP"_|_"MN"`
`:." Length of OP"=|(0-0+2)/(sqrt((1)^(2)+(-1)^(2)))|`
`=(2)/(sqrt(1+1))=(2)/(sqrt(2))=sqrt(2)`
Now, `DeltaOPN` is a right angled triangle.
So, `"ON"^(2)="OP"^(2)+"PN"^(2)" "["by Pythagoras theorem"]`
`implies" "(3)^(2)=(sqrt(2))^(2)+"PN"^(2)`
`[because" radius of circle is 3 units and OP"=sqrt(2)" units"]`
`implies" "9-2="PN"^(2)`
`implies" ""PN"^(2)=7implies"PN"=sqrt(7)`
`because" "DeltaOPN~=DeltaOPM`
`:." "MN=2PN" "[because"MP=PN]`
`=2sqrt(7)=sqrt(28)`