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प्रश्न
If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y
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उत्तर
Given: PA = PB
To Prove: 3x = 2y
Proof: Since PA = PB
So, according to the distance formula,
`=> sqrt((x-5)^2 + (y - 1)^2) = sqrt((x+1)^2 + (y - 5)^2)`
⇒(x−5)2 + (y−1)2 = (x+1)2 +(y−5)2 (Squaring both sides)
⇒x2 − 10x + 25 + y2 − 2y + 1 = x2 + 2x + 1 + y2 − 10y +25
⇒−10x − 2y = 2x − 10y
⇒8y = 12x
⇒3x = 2y
Hence, 3x = 2y.
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संबंधित प्रश्न
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. Champa and Chameli walk into the class and after observing for a few minutes, Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees.
Using distance formula, find which of them is correct.
Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
Using the distance formula, show that the given points are collinear:
(1, -1), (5, 2) and (9, 5)
`" Find the distance between the points" A ((-8)/5,2) and B (2/5,2)`
Find the relation between a and b if the point P(a ,b) is equidistant from A (6,-1) and B (5 , 8).
Find the relation between x and y if the point M (x,y) is equidistant from R (0,9) and T (14 , 11).
What point on the x-axis is equidistant from the points (7, 6) and (-3, 4)?
Calculate the distance between the points P (2, 2) and Q (5, 4) correct to three significant figures.
If the distance between point L(x, 7) and point M(1, 15) is 10, then find the value of x
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 ______.
