Advertisements
Advertisements
Question
Using distance formula decide whether the points (4, 3), (5, 1) and (1, 9) are collinear or not.
Advertisements
Solution
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
RELATED QUESTIONS
Find the distance between two points
(i) P(–6, 7) and Q(–1, –5)
(ii) R(a + b, a – b) and S(a – b, –a – b)
(iii) `A(at_1^2,2at_1)" and " B(at_2^2,2at_2)`
Find the value of x, if the distance between the points (x, – 1) and (3, 2) is 5.
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.
The value of 'a' for which of the following points A(a, 3), B (2, 1) and C(5, a) a collinear. Hence find the equation of the line.
The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 24 hours. (Use π = 3.14) ?
Determine whether the points are collinear.
P(–2, 3), Q(1, 2), R(4, 1)
Distance of point (−3, 4) from the origin is ______.
Find the relation between a and b if the point P(a ,b) is equidistant from A (6,-1) and B (5 , 8).
Prove that the following set of point is collinear :
(5 , 1),(3 , 2),(1 , 3)
In what ratio does the point P(−4, y) divides the line segment joining the points A(−6, 10) and B(3, −8)? Hence find the value of y.
A point P lies on the x-axis and another point Q lies on the y-axis.
Write the ordinate of point P.
A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.
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.
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.
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
Find distance between points O(0, 0) and B(–5, 12).
The distance of the point P(–6, 8) from the origin is ______.
Find the points on the x-axis which are at a distance of `2sqrt(5)` from the point (7, – 4). 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).
