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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
The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and R(–3, 6), find the coordinates of P.
If the point P(2, 2) is equidistant from the points A(–2, k) and B(–2k, –3), find k. Also, find the length of AP.
Find the distance between two points
(i) P(–6, 7) and Q(–1, –5)
(ii) R(a + b, a – b) and S(a – b, –a – b)
(iii) `A(at_1^2,2at_1)" and " B(at_2^2,2at_2)`
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(−3, 5), (3, 1), (0, 3), (−1, −4)
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(4, 5), (7, 6), (4, 3), (1, 2)
Show that the points A (1, –2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Using the distance formula, show that the given points are collinear:
(–2, 5), (0, 1) and (2, –3)
Find the distance between the following pair of points.
L(5, –8), M(–7, –3)
Determine whether the points are collinear.
L(–2, 3), M(1, –3), N(5, 4)
If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.
Find the distance between the following pair of point in the coordinate plane :
(5 , -2) and (1 , 5)
Find the distance between the following point :
(Sin θ - cosec θ , cos θ - cot θ) and (cos θ - cosec θ , -sin θ - cot θ)
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
Find the distance of the following points from origin.
(a+b, a-b)
The distance between the points (0, 5) and (–5, 0) is ______.
A circle drawn with origin as the centre passes through `(13/2, 0)`. The point which does not lie in the interior of the circle 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 ______.
The points (– 4, 0), (4, 0), (0, 3) are the vertices of a ______.
Point P(0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A(–1, 1) and B(3, 3).
The point A(2, 7) lies on the perpendicular bisector of line segment joining the points P(6, 5) and Q(0, – 4).
