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प्रश्न
Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (−3, 4).
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उत्तर १
Point (x, y) is equidistant from (3, 6) and (−3, 4).
∴ `sqrt((x-3)^2+(y-6)^2)= sqrt((x-(-3))^2 + (y -4)^2)`
⇒ `sqrt((x-3)^2+(y-6)^2)= sqrt((x+3)^2+(y-4)^2)`
⇒ `(x-3)^2 + (y -6)^2 = (x+3)^2 + (y-4)^2`
⇒ x2 + 9 - 6x + y2 + 36 - 12y = x2 + 9 + 6x + y2 + 16 - 8y
⇒ 36 - 16 = 6x + 6x + 12y - 8y
⇒ 20 = 12x + 4y
⇒ 3x + y = 5
⇒ 3x + y - 5 = 0
उत्तर २
The distance d between two points `(x_1, y_1)` and `(x_2, y_2)` is given by the formula
d = `sqrt((x_1-x_2)^2 + (y_1 - y_2)^2)`
Let the three given points be P(x, y), A(3, 6) and B(−3, 4).
Now let us find the distance between ‘P’ and ‘A’.
PA = `sqrt((x - 3)^2 + (y - 3))`
Now, let us find the distance between ‘P’ and ‘B’.
PB = `sqrt((x + 3)^2 + (y - 6)^2)`
It is given that both these distances are equal. So, let us equate both the above equations,
PA = PB
`sqrt((x - 3)^2 + (y - 6)^2 )`
Squaring on both sides of the equation, we get
(x - 3)2 + (y - 6)2
= (x + 3)2 + (y - 4)2
= x2 + 9 - 6x + y2 + 36 - 12y
= x2 + 9 + 6x + y2 + 16 - 8y
= 12x + 4y = 20
= 3x + y = 5
Hence, the relationship between ‘x’ and ‘y’ based on the given condition is 3x + y = 5
संबंधित प्रश्न
If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
Check whether (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.
The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 24 hours. (Use π = 3.14) ?
Determine whether the points are collinear.
P(–2, 3), Q(1, 2), R(4, 1)
Show that the points A(1, 2), B(1, 6), C(1 + 2`sqrt3`, 4) are vertices of an equilateral triangle.
Find the distance between the following point :
(sin θ , cos θ) and (cos θ , - sin θ)
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 that the following set of point is collinear :
(5 , 5),(3 , 4),(-7 , -1)
Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
Points A (-3, -2), B (-6, a), C (-3, -4) and D (0, -1) are the vertices of quadrilateral ABCD; find a if 'a' is negative and AB = CD.
Find distance between point Q(3, – 7) and point R(3, 3)
Solution: Suppose Q(x1, y1) and point R(x2, y2)
x1 = 3, y1 = – 7 and x2 = 3, y2 = 3
Using distance formula,
d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `sqrt(square - 100)`
∴ d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `square`
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 points (2, 0), (– 2, 0) and (0, 2) are vertices of a triangle. State the type of triangle with reason
Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.
The distance between the points (0, 5) and (–5, 0) is ______.
AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0). The length of its diagonal 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 y axis equidistant from B and C is ______.
What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
