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प्रश्न
Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).
उत्तर
Let the co-ordinates of the required point on y-axis be P (0, y).
The given points are A (5, 2) and B (-4, 3).
Given, PA = PB
PA2 = PB2
(0 -5)2 + (y -2)2 = (0 + 4)2 + (y - 3)2
25 + y2 + 4 - 4y = 16 + y2 + 9 - 6y
2y = -4
y = -2
Thus, the required point is (0, -2).
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संबंधित प्रश्न
Find the value of x, if the distance between the points (x, – 1) and (3, 2) is 5.
If P and Q are two points whose coordinates are (at2 ,2at) and (a/t2 , 2a/t) respectively and S is the point (a, 0). Show that `\frac{1}{SP}+\frac{1}{SQ}` is independent of t.
Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (−3, 4).
Find the co-ordinates of points of trisection of the line segment joining the point (6, –9) and the origin.
Determine whether the point is collinear.
P(–2, 3), Q(1, 2), R(4, 1)
Find the distance of the following point from the origin :
(13 , 0)
Find the point on the x-axis equidistant from the points (5,4) and (-2,3).
The centre of a circle passing through P(8, 5) is (x+l , x-4). Find the coordinates of the centre if the diameter of the circle is 20 units.
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.
Find the distance of the following points from origin.
(a cos θ, a sin θ).
Show that each of the triangles whose vertices are given below are isosceles :
(i) (8, 2), (5,-3) and (0,0)
(ii) (0,6), (-5, 3) and (3,1).
Show that the points (a, a), (-a, -a) and `(-asqrt(3), asqrt(3))` are the vertices of an equilateral triangle.
The distance between points P(–1, 1) and Q(5, –7) 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:
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?
The distance between the points A(0, 6) and B(0, –2) is ______.
The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, – 4).
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?
What is the distance of the point (– 5, 4) from the origin?
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
