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Factor them. k2 = k x k k2 - 1 = (k - 1)(k + 1) k2 - 2k + 1 = (k - 1)(k - 1) Combine the factors, eliminating duplicates. k2(k + 1)(k - 1)(k - 1) = k5 - k4 - k3 + k2, the LCM
Equation 1: y = 2x+k => y2 = 4x2+4kx+k2 when both sides are squared Equation 2: x2+y2 = 4 => y2 = 4-x2 when rearranged So by definition: 4x2+4kx+k2 = 4-x2 => 5x2+4kx+k2-4 = 0 If the values are tangent to the curve then the discriminant b2-4ac = 0 b2-4ac = 0 => 16k2-4*5*(k2-4) = 0 16k2-20k2+80 = 0 => 4k2-5k2+20 = 0 -k2+20 = 0 => -k2 = -20 Therefore the values of k are: plus or minus the square root of 20 Check: (4*k2)-(5*k2)+20 = 0
k2
k2 + 3k =108k2 + 3k - 108 = 0(k - 9) (k + 12)k = -9 , k = 12ANSWER:k = -9 , k = 12I hope this helps!
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