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प्रश्न
A circle drawn with origin as the centre passes through `(13/2, 0)`. The point which does not lie in the interior of the circle is ______.
पर्याय
`((-3)/4, 1)`
`(2, 7/3)`
`(5, (-1)/2)`
`(-6, 5/2)`
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उत्तर
A circle drawn with origin as the centre passes through `(13/2, 0)`. The point which does not lie in the interior of the circle is `underlinebb((-6, 5/2)`.
Explanation:
It is given that, centre of circle in (0, 0) and passes through the point `(13/2, 0)`.
∴ Radius of circle = Distance between (0, 0) and `(13/2, 0)`
= `sqrt((13/2 - 0)^2 + (0 - 0)^2`
= `sqrt((13/2)^2`
= `13/2`
= 6.5
A point lie outside on or inside the circles of the distance of it from the centre of the circle is greater than equal to or less than radius of the circle.
Now, to get the correct option we have to check the option one by one.
(a) Distance between (0, 0) and `((-3)/4, 1)`
= `sqrt(((-3)/4 - 0)^2 + (1 - 0)^2`
= `sqrt(9/16 + 1)`
= `sqrt(25/16)`
= `5/4`
= 1.25 < 6.5
So, the point `(-3/4, 1)` lies interior to the circle.
(b) Distance between (0, 0) and `(2, 7/3)`
= `sqrt((2 - 0)^2 + (7/3 - 0)^2`
= `sqrt(4 + 49/9)`
= `sqrt((36 + 49)/9`
= `sqrt(85/9)`
= `9.22/3`
= 3.1 < 6.5
So, the point `(2, 7/3)` lies inside the circle.
(c) Distance between (0, 0) and `(5, (-1)/2)`
= `sqrt((5 - 0)^2 + (-1/2 - 0)^2`
= `sqrt(25 + 1/4)`
= `sqrt(101/4)`
= `10.04/2`
= 5.02 < 6.5
So, the point `(5, -1/2)` lies inside the circle.
(d) Distance between (0, 0) and `(-6, 5/2)`
= `sqrt((-6 - 0)^2 + (5/2 - 0)^2`
= `sqrt(36 + 25/4)`
= `sqrt((144 + 25)/4`
= `sqrt(169/4)`
= `13/2`
= 6.5
So, the point `(-6, 5/2)` lies on the circle i.e., does not lie interior to the circle.
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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 ______.
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?
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
The points A(–1, –2), B(4, 3), C(2, 5) and D(–3, 0) in that order form a rectangle.
