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प्रश्न
Find the distance between the following pairs of points:
(−5, 7), (−1, 3)
Find the distance between the following pairs of points:
P(-5, 7), Q(-1, 3)
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उत्तर १
Distance between (−5, 7) and (−1, 3) is given by
l = `sqrt((x_2-x_1)^2+(y_2-y_1)^2)`
l = `sqrt((-5-(-1))^2 + (7 -3)^2)`
= `sqrt((-4)^2+(4)^2)`
= `sqrt(16+16) `
= `sqrt32`
= `4sqrt2` units
उत्तर २
Let the co-ordinates of point P are (x1, y1) and of point Q are (x2, y2)
P(–5, 7) = (x1, y1)
Q(–1, 3) = (x2, y2)
PQ = `sqrt((x_2-x_1)^2+(y_2-y_1)^2)` ...(By distance formula)
= `sqrt((-1-(-5))^2+(3-7)^2)`
= `sqrt((-1+5)^2+(-4)^2)`
= `sqrt(4^2+16)`
= `sqrt(16 + 16)`
= `sqrt32`
= `sqrt(16xx2)`
= `sqrt16xxsqrt2`
= 4`sqrt2` units
संबंधित प्रश्न
If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.
If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.
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)`
Find the point on the x-axis which is equidistant from (2, -5) and (-2, 9).
Find the values of x, y if the distances of the point (x, y) from (-3, 0) as well as from (3, 0) are 4.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
If the point P(x, y ) is equidistant from the points A(5, 1) and B (1, 5), prove that x = y.
Two opposite vertices of a square are (-1, 2) and (3, 2). Find the coordinates of other two
vertices.
Find the distance of the following points from the origin:
(i) A(5,- 12)
If P (x , y ) is equidistant from the points A (7,1) and B (3,5) find the relation between x and y
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)
Find the distances between the following point.
P(–6, –3), Q(–1, 9)
Show that the ▢PQRS formed by P(2, 1), Q(–1, 3), R(–5, –3) and S(–2, –5) is a rectangle.
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 distance between the following pair of point in the coordinate plane.
(1 , 3) and (3 , 9)
Find the distance of the following point from the origin :
(13 , 0)
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 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 .
Find the point on the x-axis equidistant from the points (5,4) and (-2,3).
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
The centre of a circle passing through P(8, 5) is (x+l , x-4). Find the coordinates of the centre if the diameter of the circle is 20 units.
A(-2, -3), B(-1, 0) and C(7, -6) are the vertices of a triangle. Find the circumcentre and the circumradius of the 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.
PQR is an isosceles triangle . If two of its vertices are P (2 , 0) and Q (2 , 5) , find the coordinates of R if the length of each of the two equal sides is 3.
ABC is an equilateral triangle . If the coordinates of A and B are (1 , 1) and (- 1 , -1) , find the coordinates of C.
A point P (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.
Calculate the distance between A (7, 3) and B on the x-axis whose abscissa is 11.
The distance between points P(–1, 1) and Q(5, –7) is ______
Find distance between points O(0, 0) and B(– 5, 12)
Find distance CD where C(– 3a, a), D(a, – 2a)
Seg OA is the radius of a circle with centre O. The coordinates of point A is (0, 2) then decide whether the point B(1, 2) is on the circle?
The coordinates of the point which is equidistant from the three vertices of the ΔAOB as shown in the figure is ______.
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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:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
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Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
- At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are :- A(1, 2), B(4, 3) and C(6, 6)
- Check if the Goal keeper G(–3, 5), Sweeper H(3, 1) and Wing-back K(0, 3) fall on a same straight line.
[or]
Check if the Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1) and if C is the mid-point of IJ. - If Defensive midfielder A(1, 4), Attacking midfielder B(2, –3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, find the position of E.
