Find the points on the x axis whose distances from the points a(7,6) and b 3, 4 are in the ratio 1 2

4

Let the point on x-axis be P(x, 0)

Let A be (7, 6) and B be (3, 4)

Now PA =

PA2 = (x – 7)2 + 36
and PB =

PB2 = (x – 3)2 + 16

Given PA = PB

or PA2 = PB2

or (x – 7)2 + 36 = (x – 3)2 + 16

or x2 – 14x + 49 + 36 = x2 + 9 – 6x + 16

or -14x + 6x = 25 – 85

or -8x = -60

or 8x = 60
or 2x = 15 ⇒ x =

Hence the required point is

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Let the required point on y-axis be P (0, y).

PA = `sqrt((0 - 6)^2 + (y - 7)^2)`= `sqrt(36 + y^2 + 49 - 14y)`

= `sqrt(y^2 - 14y + 85)`

PB = `sqrt((0 -4)^2 + (y + 3)^2)`= `sqrt(16 + y^2 + 9 + 6y)`

= `sqrt(y^2 + 6y + 25)`

From the given information, we have:
`"PA"/"PB" = (1)/(2)`

`"PA"^2/"PB"^2 = (1)/(4)`

`(y^2 - 14y + 85)/(y^2 + 6y + 25) = (1)/(4)`

4y2 - 56y + 340 = y2 + 6y + 25

3y2 - 62y + 315 = 0

y = `(62 ± sqrt(3844 - 3780))/(6)`

y = `(62 ± 8)/(6)`

y = `9,(35)/(3)`

Thus, the required points on y-axis are (0, 9) and `(0,(35)/(3))`.