Prove algebraically that the straight line with equation x - 2y = 10 is a tangent to the circle with equation x^2 + y^2= 20
rearrange line equation: x=10 + 2ysubstitute into circle equation: (10+2y)2 + y2=20expand: 4y2 +20y +20y + 100 + y2= 20 collect terms: 5y2 + 40y +100 = 20move all to one side: 5y2 + 40y + 80=0divide by 5: y2 + 8y + 16=0factorise: can use quadratic equation but this one is easy to spot:(y+4)2 =0therefore they meet at y= -4x= 10 + 2(-4) =2line meets the circle at (2, -4)as there is only one point of intersection, it is therefore a tangent
