Maths (locus)

a)
Circle : x² + y² + px - 8y + q = 0

Centre :
(-p/2, 4) = (-2, k)
-p/2 = -2 and k = 4
p = 4 and k = 4

Circle : x² + y² + 4x - 8y + q = 0 ...... [1]
y-axis : x = 0 ...... [2]

Put [2] into [1] :
y² - 8y + q = 0

Discriminant Δ = 0
(-8)² - 4*1*q = 0
q = 16


b)
Circle : x² + y² + 4x - 8y + 16 = 0 ...... [1]
x-axis : y = 0 ...... [3]

Put [3] into [1] :
x² + 4x + 16 = 0

Discriminant, Δ
= 4² - 4*1*16
= -40 < 0

Since Δ < 0, there is no realroot for the equation.
Hence, the circle does not cut the x-axis.

Alternative method :
Circle : x² + y² + 4x - 8y + 16 = 0

Radius of the circle
= √[(4/2)² + (-8/2)² - 16]
= 2

Distance between the centre (-2, 4) and the x-axis
= 4

Radius of the circle < Distance between the centre and the x-axis
Hence, the circle does not cut the x-axis.