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प्रश्न
Show that the point (0, 9) is equidistant from the points (–4, 1) and (4, 1).
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उत्तर
Let P(x1, y1) = P(0, 9), Q(x2, y2) = Q(–4, 1), R(x3, y3) = R(4, 1)
By distance formula,
d(P, Q) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt([(-4) - 0]^2 + (1 - 9)^2`
= `sqrt((-4)^2 + (-8)^2`
= `sqrt(16 + 64)`
= `sqrt(80)`
= `4sqrt(5)`
And
d(P, R) = `sqrt((x_3 - x_1)^2 + (y_3 - y_1)^2`
= `sqrt((4 - 0)^2 + (1 - 9)^2`
= `sqrt(4^2 + (-8)^2`
= `sqrt(16 + 64)`
= `sqrt(80)`
= `4sqrt(5)`
Here, d(P, Q) = d(P, R)
∴ The point (0, 9) is equidistant from (–4, 1) and (4, 1).
संबंधित प्रश्न
Find the distance of a point P(x, y) from the origin.
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.
Show that the points A (1, –2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
Using the distance formula, show that the given points are collinear:
(6, 9), (0, 1) and (–6, –7)
Find the distance between the following pair of points.
R(0, -3), S(0, `5/2`)
Find the distance between the following pairs of point.
W `((- 7)/2 , 4)`, X (11, 4)
Find the distance between the following point :
(Sin θ - cosec θ , cos θ - cot θ) and (cos θ - cosec θ , -sin θ - cot θ)
Find the distance of a point (7 , 5) from another point on the x - axis whose abscissa is -5.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
Find the distance between the points (a, b) and (−a, −b).
A point P (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.
Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).
The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, -1) and the length of its diameter is 20 unit.
Calculate the distance between the points P (2, 2) and Q (5, 4) correct to three significant figures.
The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.
If the distance between the points (4, P) and (1, 0) is 5, then the value of p 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 coordinates of the centroid of ΔEHJ are ______.
The distance of the point P(–6, 8) from the origin 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.
