Find the length of the chord `x^2+y^2-4y=0` along the line `x+y=1.` Also find the angle that the chord subtends at the circumference of the larger segment.

Explanation

Given circle is `x^(2)+y^(2)-4y=0` (1)
Given chord is `x+y=1`. (2)

Centre of the circle of C(0,2) and radius is CA`=2`.
AB is chord of the circle.
M is foot of perpendicular from centre on the chord AB.
Clearly, M is midpoint of AB.
Now, `CM=(|0+2-1|)/(sqrt(2))=(1)/(sqrt(2))`
Length of chord, `AB=2AM`
`=2sqrt(CA^(2)-CM^(2))=2sqrt(4-(1)/(2))=sqrt(14)`
Also, `sin alpha=(sqrt(7))/(2sqrt(2))` or `alph=sin^(-1).(sqrt(7))/(2sqrt(2))`