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प्रश्न
Find the value of a when the distance between the points (3, a) and (4, 1) is `sqrt10`
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उत्तर
The distance d between two points `(x_1,y_1)` and `(x_2, y_2)` is given by the formula
`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`
The distance between two points (3, a) and (4, 1) is given as `sqrt10`. Substituting these values in the formula for distance between two points we have
`sqrt10 = sqrt((3 - 4)^2 `+ (a - 1)^2)`
`sqrt10 = sqrt((-1)^2 + (a - 1))`
Now, squaring the above equation on both sides of the equals sign
`10 = (-1)^2 + (a - 1)^2`
`10 = 1 + (a^2 + 1 - 2a)`
`8 = a^2 - 2a`
Thus we arrive at a quadratic equation. Let us solve this now,
`a^2 - 2a - 8 = 0`
`a^2 -4a + 2a - 8 = 0`
a(a - 4) + 2(a - 4) = 0
(a - 4)(a + 2) = 0
The roots of the above quadratic equation are thus 4 and −2.
Thus the value of ‘a’ could either be 4 or -2
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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 -
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- 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:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
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Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
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[or]
Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ. - If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
