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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
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.
Show that the points (a, a), (–a, –a) and (– √3 a, √3 a) are the vertices of an equilateral triangle. Also find its area.
Find the distance between the following pairs of points:
(2, 3), (4, 1)
Find the values of x, y if the distances of the point (x, y) from (-3, 0) as well as from (3, 0) are 4.
Using the distance formula, show that the given points are collinear:
(1, -1), (5, 2) and (9, 5)
Find the distance between the following pair of point.
P(–5, 7), Q(–1, 3)
Show that the points A(1, 2), B(1, 6), C(1 + 2`sqrt3`, 4) are vertices of an equilateral triangle.
Find the distance of the following point from the origin :
(5 , 12)
Find the distance between P and Q if P lies on the y - axis and has an ordinate 5 while Q lies on the x - axis and has an abscissa 12 .
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.
Given A = (3, 1) and B = (0, y - 1). Find y if AB = 5.
If the length of the segment joining point L(x, 7) and point M(1, 15) is 10 cm, then the value of x is ______.
Show that P(–2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle.
The distance between the point P(1, 4) and Q(4, 0) 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?
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 ______.
If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.
The point P(–2, 4) lies on a circle of radius 6 and centre C(3, 5).
Find the distance between the points O(0, 0) and P(3, 4).
