For the circle , AC and CB are orthogonal
According to the question ,
AB is the diameter of a circle with center O and the C is a point on one
of the two arcs joining A and B
Let AO = -u and OB = u and CO = v
To prove that AC and AB is orthogonal .
We have to show their dot product is zero i.e. CA . CB = 0
=> CO + CA = -u
=> CA = -v -u
=> CO + CB = -v
=> CB = -v + u
So, the dot product,
CA . CB = (-v -u ) . ( -v +u)
=> v .v + u . v - v . u - u .u
=> |v|² +u . v - u . v - |u|²
=> |v|² - |u|² = 0
Hence , the length of v is same as the length of u
Therefore , AC and AB are orthogonal
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