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Question
Find distance CD where C(– 3a, a), D(a, – 2a)
Solution
Let C(x1, y1) and D(x2, y2) be the given points
∴ x1 = – 3a, y1 = a, x2 = a, y2 = – 2a
By distance formula,
d(C, D) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt(["a" - (-3"a")]^2 + (-2"a" - "a")^2`
= `sqrt(("a" + 3"a")^2 + (-2"a" - "a")^2`
= `sqrt((4"a")^2 + (-3"a")^2`
= `sqrt(16"a"^2 + 9"a"^2)`
= `sqrt(25"a"^2)`
∴ d(C, D) = 5a units
RELATED QUESTIONS
If A(5, 2), B(2, −2) and C(−2, t) are the vertices of a right angled triangle with ∠B = 90°, then find the value of t.
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:
(a, b), (−a, −b)
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.
Find the distance of a point P(x, y) from the origin.
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.
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 point :
(sec θ , tan θ) and (- tan θ , sec θ)
Find the value of m if the distance between the points (m , -4) and (3 , 2) is 3`sqrt 5` units.
ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
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.
Calculate the distance between A (5, -3) and B on the y-axis whose ordinate is 9.
The distance between point P(2, 2) and Q(5, x) is 5 cm, then the value of x ______
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 ______
AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0). The length of its diagonal 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:
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 ______.
Find distance between points P(– 5, – 7) and Q(0, 3).
By distance formula,
PQ = `sqrt(square + (y_2 - y_1)^2`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(square + square)`
= `sqrt(125)`
= `5sqrt(5)`
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
