Is line y=kx+b tangent to circle x^2+y^2=1?

Since the line cuts the circle at only one point, roots of the \(x^2 + (kx+b)^2 =1\) must be equal. {complex roots of a quadratic equation always occur in pair and are complementary, hence if you have one real root, other must be real.}

Red line doesn't cut the circle. For this line, discriminant of the equation is negative. (equation doesn't have any real root)

Brown line cuts the circle at 2 distinct points. For this line, discriminant of the equation is positive. (equation has 2 distinct roots or 2 distinct values of x)

Blue line (line given in our question) cuts the circle at one distinct point. For this line, discriminant of the equation is zero. (equation has 2 equal roots or 1 value of x)

If you still have any doubt, you can ask.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.