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Question
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
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Solution
Let P(h, k) be the point which is equidistant from the points A(–5, 4) and B(–1, 6).
∴ PA = PB ...`[∵ "By distance formula, distance" = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`
⇒ (PA)2 = (PB)2
⇒ (– 5 – h)2 + (4 – k)2 = (– 1 – h)2 + (6 – k)2
⇒ 25 + h2 + 10h + 16 + k2 – 8k = 1 + h2 + 2h + 36 + k2 – 12k
⇒ 25 + 10h + 16 – 8k = 1 + 2h + 36 – 12k
⇒ 8h + 4k + 41 – 37 = 0
⇒ 8h + 4k + 4 = 0
⇒ 2h + k + 1 = 0 ...(i)
Mid-point of AB = `((-5 - 1)/2, (4 + 6)/2)` = (– 3, 5) ...`[∵ "Mid-point" = ((x_1 + x_2)/2, (y_1 + y_2)/2)]`
At point (– 3, 5), from equation (i),
2h + k = 2(– 3) + 5
= – 6 + 5
= – 1
⇒ 2h + k + 1 = 0
So, the mid-point of AB satisfy the equation (i).
Hence, infinite number of points, in fact all points which are solution of the equation 2h + k + 1 = 0, are equidistant from the points A and B.
Replacing h, k by x, y in above equation, we have 2x + y + 1 = 0
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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:
The coordinates of the centroid of ΔEHJ are ______.
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:
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By distance formula,
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= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(125)`
= `5sqrt(5)`
