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प्रश्न
Find the coordinates of the points on the y-axis, which are at a distance of 10 units from the point (-8, 4).
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उत्तर
Let the coordinates of the point on y-axis be (0, y).
From the given information, we have:
`sqrt((0 + 8)^2 + (y - 4)^2)` = 10
(0 + 8)2 + (y - 4)2 = 100
64 + y2 + 16 - 8y = 100
y2 - 8y - 20 = 0
y2 - 10y + 2y - 20 = 0
y(y - 10) + 2(y - 10) = 0
(y - 10)(y + 2) = 0
y = 10, - 2
Thus, the required co-ordinates of the points on y-axis are (0, 10) and (0, -2).
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संबंधित प्रश्न
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Using the distance formula, show that the given points are collinear:
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Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.
Use distance formula to show that the points A(-1, 2), B(2, 5) and C(-5, -2) are collinear.
Find distance between point A(–1, 1) and point B(5, –7):
Solution: Suppose A(x1, y1) and B(x2, y2)
x1 = –1, y1 = 1 and x2 = 5, y2 = – 7
Using distance formula,
d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ d(A, B) = `sqrt(square +[(-7) + square]^2`
∴ d(A, B) = `sqrt(square)`
∴ d(A, B) = `square`
Show that A(1, 2), (1, 6), C(1 + 2 `sqrt(3)`, 4) are vertices of a equilateral triangle
The equation of the perpendicular bisector of line segment joining points A(4,5) and B(-2,3) is ______.
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?
Points A(4, 3), B(6, 4), C(5, –6) and D(–3, 5) are the vertices of a parallelogram.
