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प्रश्न
Using distance formula decide whether the points (4, 3), (5, 1) and (1, 9) are collinear or not.
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उत्तर
Let A(x1, y1) = A(4, 3), B(x2, y2) = B(5, 1), C(x3, y3) = C(1, 9)
∴ d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt((5 -4)^2 + (1 - 3)^2`
= `sqrt(1^2 + (-2)^2`
= `sqrt(1+ 4)`
= `sqrt(5)` ...(i)
∴ d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2`
= `sqrt((1 - 5)^2 + (9 - 1)^2`
= `sqrt((-4)^2 + 8^2`
= `sqrt(16 + 64)`
=`sqrt(80)`
= `4sqrt(5)` ...(ii)
∴ d(A, C) = `sqrt((x_3 -x_1)^2 + (y_3 - y_2)^2`
= `sqrt((1 - 4)^2 + (9 - 3)^2`
= `sqrt((-3)^2 + 6^2`
= `sqrt(9 + 36)`
= `sqrt(45)`
= `3sqrt(5)` ...(iii)
`sqrt(5) + 3sqrt(5) = 4sqrt(5)`
∴ d(A, B) + d(A, C) = d(B, C) ...[From (i), (ii) and (iii)]
∴ Points A, B, C are collinear.
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संबंधित प्रश्न
If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y
Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
If the points (2, 1) and (1, -2) are equidistant from the point (x, y), show that x + 3y = 0.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.
Find the distance of the following point from the origin :
(5 , 12)
Find the distance of a point (7 , 5) from another point on the x - axis whose abscissa is -5.
Find the relation between x and y if the point M (x,y) is equidistant from R (0,9) and T (14 , 11).
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
A line segment of length 10 units has one end at A (-4 , 3). If the ordinate of te othyer end B is 9 , find the abscissa of this end.
The centre of a circle passing through P(8, 5) is (x+l , x-4). Find the coordinates of the centre if the diameter of the circle is 20 units.
The distance between the points (3, 1) and (0, x) is 5. Find x.
A point A is at a distance of `sqrt(10)` unit from the point (4, 3). Find the co-ordinates of point A, if its ordinate is twice its abscissa.
The distance between point P(2, 2) and Q(5, x) is 5 cm, then the value of x = ______.
Find distance of point A(6, 8) from origin.
The coordinates of the point which is equidistant from the three vertices of the ΔAOB as shown in the figure 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:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
The distance between the points A(0, 6) and B(0, –2) is ______.
What is the distance of the point (– 5, 4) from the origin?
