F4 Maths

Given a point P(m, n), its perpendicular distance from the x-axis and its distance from the point (4,2) are 5 units. Find P(m,n).


The perpendicular distance of P(m, n) from the axis:
n = 5 or n = -5

When n = 5:
The point is P(m, 5).
Distance between P and the point (4, 2):
√[(m- 4)² + (5 - 2)²] = 5
(m- 4)² + (5 - 2)² = 25
m² - 8m + 16 + 9 = 25
m² - 8m = 0
m(m - 8) = 0
m = 0 or m = 80

When n = -5:
The point is P(m, -5).
Distance between P and the point (4, 2):
√[(m- 4)² + (-5 - 2)²] = 5
(m- 4)² + (-5 - 2)² = 25
m² - 8m + 16 + 49 = 25
m² - 8m + 40 = 0
Determinant Δ= (-8)² - 4(40) < 0
There is no real roots.

Hence, P(m, n) = P(0, 5) or P(8, 5)