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प्रश्न
The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, – 9) and has diameter `10sqrt(2)` units.
उत्तर
By given condition,
Distance between the centre C(2a, a – 7) and the point P(11, – 9), which lie on the circle = Radius of circle
∴ Radius of circle = `sqrt((11 - 2a)^2 + (-9 - a + 7)^2` ...(i) `[∵ "Distance between two points" (x_1, y_1) "and" (x_2, y_2) = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`
Given that, length of diameter = `10sqrt(2)`
∴ Length of radius = `"Length of diameter"/2`
= `(10sqrt(2))/2`
= `5sqrt(2)`
Put this value in equation (i), we get
`5sqrt(2) = sqrt((11 - 2a)^2 + (-2 - a)^2`
Squaring on both sides, we get
50 = (11 – 2a)2 + (2 + a)2
⇒ 50 = 121 + 4a2 – 44a + 4 + a2 + 4a
⇒ 5a2 – 40a + 75 = 0
⇒ a2 – 8a + 15 = 0
⇒ a2 – 5a – 3a + 15 = 0 ...[By fractorisation method]
⇒ a(a – 5) – 3(a – 5) = 0
⇒ (a – 5)(a – 3) = 0
∴ a = 3, 5
Hence, the required values of a are 5 and 3.
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संबंधित प्रश्न
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 distance between the points
A(-6,-4) and B(9,-12)
Find the value of y for which the distance between the points A (3, −1) and B (11, y) is 10 units.
Find the distance between the following pairs of point:
`(sqrt(3)+1,1)` and `(0, sqrt(3))`
What point on the x-axis is equidistant from the points (7, 6) and (-3, 4)?
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
Show that P(– 2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle
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 ______.
A circle has its centre at the origin and a point P(5, 0) lies on it. The point Q(6, 8) lies outside the circle.
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
