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प्रश्न
Find the point on the x-axis which is equidistant from (2, -5) and (-2, 9).
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उत्तर १
We have to find a point on x-axis. Therefore, its y-coordinate will be 0.
Let the point on x-axis be (x, 0)
Distance between (x, 0) and (2, -5) = `sqrt((x-2)^2+(0-(-5))^2)`
= `sqrt((x-2)^2+(5)^2)`
Distance between (x, 0) and (-2, -9) = `sqrt((x-(-2))^2+(0-(-9))^2)`
= `sqrt((x+2)^2+(9)^2)`
By the given condition, these distances are equal in measure.
`sqrt((x-2)^2 +(5)^2)`
= `sqrt((x+2)^2+(9)^2)`
= (x - 2)2 + 25 = (x + 2)2 + 81
= x2 + 4 - 4x + 25
= x2 + 4 + 4x + 81
8x = 25 - 81
8x = -56
x = -7
Therefore, the point is (−7, 0).
उत्तर २
Let (x, 0) be the point on the x axis. Then as per the question, we have
⇒ `sqrt((x-2)^2 +(0+5)^2)`
⇒ `sqrt((x+2)^2 + (0-9)^2)`
⇒ `sqrt((x-2)^2 +(5)^2)=sqrt((x+2)^2 + (9)^2)`
⇒ (x - 2)2 + (5)2 = (x + 2)2 + (-9)2 ...(Squaring both sides)
⇒ x2 - 4x + 4 + 25 = x2 + 4x + 4 + 81
8x = 25 - 81
8x = -56
x = -7
Therefore, the point is (−7, 0).
संबंधित प्रश्न
Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.
Find the distance between the following pair of points:
(a+b, b+c) and (a-b, c-b)
If the points (2, 1) and (1, -2) are equidistant from the point (x, y), show that x + 3y = 0.
Find all possible values of x for which the distance between the points
A(x,-1) and B(5,3) is 5 units.
Find x if distance between points L(x, 7) and M(1, 15) is 10.
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
PQR is an isosceles triangle . If two of its vertices are P (2 , 0) and Q (2 , 5) , find the coordinates of R if the length of each of the two equal sides is 3.
ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.
The distance between the points (3, 1) and (0, x) is 5. Find x.
Show that (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus.
Point P (2, -7) is the centre of a circle with radius 13 unit, PT is perpendicular to chord AB and T = (-2, -4); calculate the length of AB.
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) 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:
The point on x axis equidistant from I and E is ______.
The points A(–1, –2), B(4, 3), C(2, 5) and D(–3, 0) in that order form a rectangle.
|
Case Study Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites. |
- Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.
- After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by 240(`sqrt(3)` - 1) m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?
Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
|
Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
- At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1, 2), B(4, 3) and C(6, 6)
- Check if the Goal keeper G(–3, 5), Sweeper H(3, 1) and Wing-back K(0, 3) fall on a same straight line.
[or]
Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ. - If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
