Advertisements
Advertisements
प्रश्न
Using distance formula decide whether the points (4, 3), (5, 1) and (1, 9) are collinear or not.
Advertisements
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.
APPEARS IN
संबंधित प्रश्न
If the point P(2, 2) is equidistant from the points A(–2, k) and B(–2k, –3), find k. Also, find the length of AP.
Find the distance between the following pairs of points:
(2, 3), (4, 1)
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. Champa and Chameli walk into the class and after observing for a few minutes, Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees.
Using distance formula, find which of them is correct.
ABC is a triangle and G(4, 3) is the centroid of the triangle. If A = (1, 3), B = (4, b) and C = (a, 1), find ‘a’ and ‘b’. Find the length of side BC.
Find the distance between the following pair of points:
(a sin α, –b cos α) and (–a cos α, b sin α)
If P(x, y) is equidistant from the points A(7, 1) and B(3, 5), find the relation between x and y.
For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?
Find the distances between the following point.
R(–3a, a), S(a, –2a)
Find the distance between the following pairs of point in the coordinate plane :
(13 , 7) and (4 , -5)
Find the value of a if the distance between the points (5 , a) and (1 , 5) is 5 units .
Prove that the following set of point is collinear :
(4, -5),(1 , 1),(-2 , 7)
Prove that the points (0 , 0) , (3 , 2) , (7 , 7) and (4 , 5) are the vertices of a parallelogram.
KM is a straight line of 13 units If K has the coordinate (2, 5) and M has the coordinates (x, – 7) find the possible value of x.
Show that the points (2, 0), (–2, 0) and (0, 2) are vertices of a triangle. State the type of triangle with reason.
Show that A(1, 2), (1, 6), C(1 + 2`sqrt(3)`, 4) are vertices of an equilateral triangle.
The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) is ______.
If the distance between the points (x, -1) and (3, 2) is 5, then the value of x 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:
The coordinates of the centroid of ΔEHJ are ______.
Point P(0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A(–1, 1) and B(3, 3).
