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