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Question
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
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Solution 1
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))`.
Solution 2
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))`.
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Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
The points A(–1, –2), B(4, 3), C(2, 5) and D(–3, 0) in that order form a rectangle.
