Advertisements
Advertisements
Question
If (a, b) is the mid-point of the line segment joining the points A(10, –6) and B(k, 4) and a – 2b = 18, find the value of k and the distance AB.
Advertisements
Solution
Since, (a, b) is the mid-point of line segment AB.
∴ (a, b) = `((10 + "k")/2, (-6 + 4)/2)` ...`["Since, mid-point of a line segment having points" (x_1, y_1) "and" (x_2, y_2) = ((x_1 + x_2)/2, (y_1 + y_2)/2)]`
⇒ (a, b) = `((10 + "k")/2, -1)`
Now, equating coordinates on both sides, we get
∴ a = `(10 + "k")/2` ...(i)
And b = –1 ...(ii)
Given, a – 2b = 18
From equation (ii),
a – 2(–1) = 18
⇒ a + 2 = 18
⇒ a = 16
From equation (i),
16 = `(10 + "k")/2`
⇒ 32 = 10 + k
⇒ k = 22
Hence, the required value of k is 22.
⇒ k = 22
∴ A = (10 – 6), B = (22, 4)
Now, distance between A(10, –6) and B(22, 4),
AB = `sqrt((22 - 10)^2 + (4 + 6)^2` ...`[∵ "Distance between the point" (x_1, y_1) "and" (x_2, y_2), d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`
= `sqrt((12)^2 + (10)^2`
= `sqrt(144 + 100)`
= `sqrt(244)`
= `2sqrt(61)`
Hence, the required distance of AB is `2sqrt(61)`.
APPEARS IN
RELATED QUESTIONS
Check whether (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.
Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.
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.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
The long and short hands of a clock are 6 cm and 4 cm long respectively. Find the sum of the distances travelled by their tips in 24 hours. (Use π = 3.14) ?
Determine whether the points are collinear.
P(–2, 3), Q(1, 2), R(4, 1)
Find the distance between the following pairs of point in the coordinate plane :
(7 , -7) and (2 , 5)
Find the distance of the following point from the origin :
(0 , 11)
Prove that the points (1 , 1) , (-1 , -1) and (`- sqrt 3 , sqrt 3`) are the vertices of an equilateral triangle.
Prove that the points (0 , 2) , (1 , 1) , (4 , 4) and (3 , 5) are the vertices of a rectangle.
From the given number line, find d(A, B):
Show that (-3, 2), (-5, -5), (2, -3) and (4, 4) are the vertices of a rhombus.
Find the distance of the following points from origin.
(5, 6)
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
If the distance between point L(x, 7) and point M(1, 15) is 10, then find the value of x.
Show that the point (11, –2) is equidistant from (4, –3) and (6, 3).
The distance between the points A(0, 6) and B(0, -2) is ______.
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
|
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.
|
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).
The distance of the point (5, 0) from the origin is ______.
