A. at what points will the line y = x intersect the unit circle x2 + y2 = 1? answer (1/√2, 1/√2), (−1/√2,−1/√2)

b. at what points will the line y = −x intersect the unit circle x2 + y2 = 1? answer (1/√2, -1/√2), (−1/√2,1/√2)

A.    In circle, x^2+y^2 = 1

given line x=y

put in unit circle,

x^2+(x)^2=1

2x^2=1

x^2= 1/2

x = sqrt [1/2] = y       ,      x = - sqrt [1/2] = y

( sqrt [1/2] , sqrt [1/2] )    ,   ( -sqrt [1/2] , -sqrt [1/2] )

Line x=y intersects the unit circle at both the points.

B.   In circle, x^2+y^2 = 1

given line y = - x

put in unit circle,

x^2+(-x)^2=1

2x^2=1

x^2= 1/2

x = sqrt [1/2]        ,      x = - sqrt [1/2]

y= -sqrt [1/2]        ,      y =   sqrt [1/2]

( sqrt [1/2] , -sqrt [1/2] )    ,   ( -sqrt [1/2] , sqrt [1/2] )

Line y = -x intersects the unit circle at both the points.