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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
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 value of x, if the distance between the points (x, – 1) and (3, 2) is 5.
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)
If Q (0, 1) is equidistant from P (5, –3) and R (x, 6), find the values of x. Also, find the distances QR and PR.
ABC is a triangle and G(4, 3) is the centroid of the triangle. If A = (1, 3), B = (4, b) and C = (a, 1), find ‘a’ and ‘b’. Find the length of side BC.
Find the distance between the following pair of points:
(a + b, b + c) and (a – b, c – b)
Find the value of a when the distance between the points (3, a) and (4, 1) is `sqrt10`.
An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.
Find the distance of the following points from the origin:
C(–4, –6)
Find all possible values of x for which the distance between the points A(x, –1) and B(5, 3) is 5 units.
Using the distance formula, show that the given points are collinear:
(–1, –1), (2, 3) and (8, 11)
Find the distance between the following pairs of point.
W `((- 7)/2 , 4)`, X (11, 4)
Find the distance between the following pair of point in the coordinate plane :
(5 , -2) and (1 , 5)
Find the distance between the following pairs of point in the coordinate plane :
(13 , 7) and (4 , -5)
Find the relation between a and b if the point P(a ,b) is equidistant from A (6,-1) and B (5 , 8).
Prove that the following set of point is collinear :
(5 , 5),(3 , 4),(-7 , -1)
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles 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.
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.
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.
Calculate the distance between A (7, 3) and B on the x-axis whose abscissa is 11.
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Find distance between point A(7, 5) and B(2, 5).
Find distance between points O(0, 0) and B(–5, 12).
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 CD where C(–3a, a), D(a, –2a).
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 distance of the point P(–6, 8) from the origin is ______.
Name the type of triangle formed by the points A(–5, 6), B(–4, –2) and C(7, 5).
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
The centre of a circle is (2a, a – 7). Find the values of a if the circle passes through the point (11, –9) and has diameter `10sqrt(2)` units.
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Case Study Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites. |
- Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.
- After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by 240(`sqrt(3)` - 1) m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?
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In a GPS, The lines that run east-west are known as lines of latitude, and the lines running north-south are known as lines of longitude. The latitude and the longitude of a place are its coordinates and the distance formula is used to find the distance between two places. The distance between two parallel lines is approximately 150 km. A family from Uttar Pradesh planned a round trip from Lucknow (L) to Puri (P) via Bhuj (B) and Nashik (N) as shown in the given figure below.
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Based on the above information answer the following questions using the coordinate geometry.
- Find the distance between Lucknow (L) to Bhuj (B).
- If Kota (K), internally divide the line segment joining Lucknow (L) to Bhuj (B) into 3 : 2 then find the coordinate of Kota (K).
- Name the type of triangle formed by the places Lucknow (L), Nashik (N) and Puri (P)
[OR]
Find a place (point) on the longitude (y-axis) which is equidistant from the points Lucknow (L) and Puri (P).
