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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.
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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
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.
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Find the value of a when the distance between the points (3, a) and (4, 1) is `sqrt10`
Find all possible values of y for which distance between the points is 10 units.
Find the distance between the following pair of points.
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Find the distance between the following pairs of point.
W `((- 7)/2 , 4)`, X (11, 4)
Find the distances between the following point.
R(–3a, a), S(a, –2a)
Find the distance of the following point from the origin :
(5 , 12)
Find the distance of the following point from the origin :
(13 , 0)
P(5 , -8) , Q (2 , -9) and R(2 , 1) are the vertices of a triangle. Find tyhe circumcentre and the circumradius of the triangle.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Prove that the points (1 , 1) , (-1 , -1) and (`- sqrt 3 , sqrt 3`) are the vertices of an equilateral triangle.
Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.
The distance between the points (3, 1) and (0, x) is 5. Find x.
Given A = (3, 1) and B = (0, y - 1). Find y if AB = 5.
Given A = (x + 2, -2) and B (11, 6). Find x if AB = 17.
Show that the points (2, 0), (–2, 0) and (0, 2) are vertices of a triangle. State the type of triangle with reason.
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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:
The point on y axis equidistant from B and C is ______.
