For what value of b is the line y = bx + 3 a tangent to x^2 + y^2 = 36

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A line is a tangent to a circle if it intersects the circle at only one point. The value of b has to be determined that makes the line y = bx + 3 a tangent to x^2 + y^2 = 36.

Substituting y = bx + 3 in x^2 + y^2 = 36 gives

x^2 + (bx + 3)^2 = 36

=> x^2 + b^2x^2 + 6bx + 9 = 36

=> x^2(1 + b^2) + 6bx - 27 = 0

For the quadratic equation derived to have one root

(6b)^2 = 4*(-27)(1 + b^2)

=> 36b^2 = 108 - 108b^2

=> 144b^2 = 108

=> b^2 = 108/144 = 3/4

=> b = `sqrt 3/2` and b = `-sqrt 3/2`

**For b = `sqrt 3/2` and `-sqrt 3/2` , y = bx + 3 a tangent to x^2 + y^2 = 36.**

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