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प्रश्न
If the point A(x,2) is equidistant form the points B(8,-2) and C(2,-2) , find the value of x. Also, find the value of x . Also, find the length of AB.
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उत्तर
As per the question
AB = AC
`⇒ sqrt((x-8)^2+(2+2)^2 ) = sqrt((x-2)^2 +(2+2)^2)`
Squaring both sides, we get
`(x-8)^2 +4^2 = (x - 2)^2 +4^2`
`⇒ x^2 -16x+64+16=x^2+4-4x+16`
`⇒ 16x-4x=64-4`
`⇒ x = 60/12=5`
Now,
`AB = sqrt((x-8)^2 +(2+2)^2)`
`= sqrt((5-8)^2 +(2+2)^2) (∵ x =2)`
`=sqrt((-3)^2 +(4)^2)`
`=sqrt(9+16) = sqrt(25)=5`
Hence, x = 5and AB = 5 units.
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संबंधित प्रश्न
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(4, 5), (7, 6), (4, 3), (1, 2)
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(asinα, −bcosα) and (−acos α, bsin α)
If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?
Determine whether the points are collinear.
L(–2, 3), M(1, –3), N(5, 4)
Distance of point (−3, 4) from the origin is ______.
Find the distance between the following pairs of point in the coordinate plane :
(4 , 1) and (-4 , 5)
Find the distance of the following point from the origin :
(0 , 11)
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Show that the points (a, a), (-a, -a) and `(-asqrt(3), asqrt(3))` are the vertices of an equilateral triangle.
If the length of the segment joining point L(x, 7) and point M(1, 15) is 10 cm, then the value of x is ______
The distance between the points A(0, 6) and B(0, -2) is ______.
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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:
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?
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.
Find the distance between the points O(0, 0) and P(3, 4).
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
