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प्रश्न
The distance between point P(2, 2) and Q(5, x) is 5 cm, then the value of x = ______.
विकल्प
2
6
3
1
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उत्तर
The distance between point P(2, 2) and Q(5, x) is 5 cm, then the value of x = 6.
Explanation:
Let P(x1, y1) = P(2, 2) and Q(x2, y2) = Q(5, x)
Here, x1 = 2, y1 = 2, x2 = 5, y2 = x
By distance formula,
d(P, Q) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
∴ `5 = sqrt((5 - 2)^2 + (x - 2)^2)`
∴ `5 = sqrt(9 + x^2 - 4x + 4)`
∴ 52 = x2 – 4x + 13 ...[Squaring both sides]
∴ 25 = x2 – 4x + 13
∴ x2 – 4x + 13 – 25 = 0
∴ x2 – 4x – 12 = 0
∴ (x – 6) (x + 2) = 0
∴ x – 6 = 0 or x + 2 = 0
∴ x = 6 or x = –2
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संबंधित प्रश्न
If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.
If Q (0, 1) is equidistant from P (5, –3) and R (x, 6), find the values of x. Also, find the distances QR and PR.
Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (–3, 4).
If the points (2, 1) and (1, -2) are equidistant from the point (x, y), show that x + 3y = 0.
AB and AC are the two chords of a circle whose radius is r. If p and q are
the distance of chord AB and CD, from the centre respectively and if
AB = 2AC then proove that 4q2 = p2 + 3r2.
Find the distance of the following point from the origin :
(8 , 15)
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
The distance between the points (3, 1) and (0, x) is 5. Find x.
Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.
Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square ABCD.
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).
Find distance between point A(–1, 1) and point B(5, –7):
Solution: Suppose A(x1, y1) and B(x2, y2)
x1 = –1, y1 = 1 and x2 = 5, y2 = –7
Using distance formula,
d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ d(A, B) = `sqrt(square +[(-7) + square]^2`
∴ d(A, B) = `sqrt(square)`
∴ d(A, B) = `square`
Show that the point (0, 9) is equidistant from the points (–4, 1) and (4, 1).
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:
The point on x axis equidistant from I and E is ______.
Points A(4, 3), B(6, 4), C(5, –6) and D(–3, 5) are the vertices of a parallelogram.
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
The distance between the points (0, 5) and (–3, 1) is ______.
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
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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.
