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Question
Find the distance between the following pair of points:
(asinα, −bcosα) and (−acos α, bsin α)
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Solution
The distance d between two points (x1, y1) and (x2, y2) is given by the formula.
`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`
The two given points are (asinα, −bcosα) and (−acos α, bsin α)
The distance between these two points is
`d = sqrt((a sin alpha + a cos alpha)^2 + (- b cos alpha - bsin alpha)^2)`
`= sqrt(a^2(sin alpha + cos alpha)^2 + b^2(-1)^2(cos alpha + sin alpha))`
`= sqrt(a^2(sin alpha + cos alpha)^2 + b^2(sin alpha + cos alpha))`
`= sqrt((a^2 + b^2)(sin alpha + cos alpha))`
`d = (sin alpha + cos alpha)sqrt(a^2 + b^2)`
Hence the distance is `(sin alpha + cos alpha)sqrt((a^2 + b^2))`
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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?
What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?
Ayush starts walking from his house to office. Instead of going to the office directly, he goes to a bank first, from there to his daughter’s school and then reaches the office. What is the extra distance travelled by Ayush in reaching his office? (Assume that all distances covered are in straight lines). If the house is situated at (2, 4), bank at (5, 8), school at (13, 14) and office at (13, 26) and coordinates are in km.
