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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