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प्रश्न
If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.
If a point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), then find the value of p.
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उत्तर १
The given points are A(0, 2), B(3, p) and C(p, 5).
It is given that A is equidistant from B and C.
∴ AB = AC
⇒ AB2 = AC2
⇒ (3 − 0)2 + (p − 2)2 = (p − 0)2 + (5 − 2)2
⇒ 9 + p2 + 4 − 4p = p2 + 9
⇒ 4 − 4p = 0
⇒ 4p = 4
⇒ p = 1
Thus, the value of p is 1.
Length of AB `=sqrt((3-0)^2+(1-2)^2)=sqrt(3^2+(-1)^2)=sqrt(9+1)=sqrt(10) units`
उत्तर २
It is given that A(0, 2) is equidistant from the points B(3, p) and C(p, 5).
∴ AB = AC
\[\Rightarrow \sqrt{\left( 3 - 0 \right)^2 + \left( p - 2 \right)^2} = \sqrt{\left( p - 0 \right)^2 + \left( 5 - 2 \right)^2}\] (Distance formula)
Squaring on both sides, we get
\[9 + p^2 - 4p + 4 = p^2 + 9\]
\[ \Rightarrow - 4p + 4 = 0\]
\[ \Rightarrow p = 1\]
Thus, the value of p is 1.
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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.
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- 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 point on y axis equidistant from B and C is ______.
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