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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
If a ≠ b ≠ 0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
The value of 'a' for which of the following points A(a, 3), B (2, 1) and C(5, a) a collinear. Hence find the equation of the line.
Find the value of a when the distance between the points (3, a) and (4, 1) is `sqrt10`.
Find the distance of the following points from the origin:
B(–5, 5)
Find all possible values of y for which the distance between the points A(2, –3) and B(10, y) is 10 units.
Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.
Find the distance between the following pair of points.
R(0, -3), S(0, `5/2`)
Find the distances between the following point.
R(–3a, a), S(a, –2a)
If A and B are the points (−6, 7) and (−1, −5) respectively, then the distance
2AB is equal to
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 coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram.
A point P lies on the x-axis and another point Q lies on the y-axis.
Write the abscissa of point Q.
The length of line PQ is 10 units and the co-ordinates of P are (2, -3); calculate the co-ordinates of point Q, if its abscissa is 10.
Find the distance of the following points from origin.
(a+b, a-b)
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 ______.
The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, – 4).
The distance between the points (0, 5) and (–3, 1) is ______.
