Answer:
k = 1/2
Step-by-step explanation:
input y = kx +6 into the equation of the curve x² + y² – 10x + 8y = 84
x² + (kx + 6)² - 10x + 8(kx + 6) = 84
expand:
x² + k²x² + 12kx + 36 - 10x + 8kx + 48 = 84
simplify by collecting like terms:
x² + k²x² + 20kx - 10x + 84 = 84
subtract 84 on both sides to bring it to the left:
x² + k²x² + 20kx - 10x + 84 - 84 = 0
x² + k²x² + 20kx - 10x = 0
factorise out x:
x²(1 + k²) + x(20k - 10) = 0
using the discriminant b² - 4ac where b is 20k - 10, a is 1 + k² and c is 0, substitute them in the formula b² - 4ac:
b² - 4ac
(20k - 10)² - 4(1 + k²)(0) = 0
the part highlighted in bold is gone because it's all multiplied by 0, so we are left with (20k - 10)² = 0
(20k - 10)² is the same as
(20k - 10)(20k - 10)
equate both to 0
20k - 10 = 0 and 20k - 10 = 0
add 10 on both sides
20k = 10 and 20k = 10
divide 20 on both sides
k = 10/20 and k = 10/20 which are both the same
10/20 is simplified to 1/2
k = 1/2
