Question

Find the points on the locus of points that are 3 units from x-axis and 5 units from the point (5, 1)

Answer 1

Given that the required point is 3 units from x-axis and 5 units from the point P(5, 1).

Let Q(h, 3) and K(h, – 3) be the required points.

∴ PQ = 5

`sqrt((5 -"h")^2 + (1 - 3)^2` = 5

(5 – h)² + (– 2)² = 25

25 – 10h + h² + 4 = 25

h² – 10h + 29 – 25 = 0

h² – 10h + 4 = 0

h = `(10 +-  sqrt((10))^2 - 4 xx 1 xx 4)/2`

h = `(10 +- sqrt(100 - 16))/2`

= `(10 +- sqrt(84))/2`

= `(10 +-  sqrt( xx 21))/2`

= `(10 +- 2sqrt(21))/2`

= `5 +-  sqrt(21)`

∴ `"Q"(5 + sqrt(21), 3), (5 - sqrt(21), 3)`
PR = 5

∴ `sqrt((5 -"k")^2 + (1 + 3)^2` = 5

(5 – k)² + 4² = 25

25 – 10k + k² + 16 = 25

k² – 10k + 16 = 0

k = `(10 +-  sqrt((- 10))^2 - 4 xx 16)/2`

k = `(10 +-  sqrt(100 - 64))/2` 

= `(10 +-  sqrt(36))/2`

k = `(10 +- 6)/2`

k = `(10 + 6)/2` or k = `(10 - 6)/2`

k =  8 or k = 2

R = (8, – 3), (2, – 3)

∴ Required points are `(5 + sqrt(21), 3)(5 -sqrt(21), 3), (8 - 3), (2 - 3)`