Advertisements
Advertisements
प्रश्न
Find a point which is equidistant from the points A(–5, 4) and B(–1, 6)? How many such points are there?
Advertisements
उत्तर
Let P(h, k) be the point which is equidistant from the points A(–5, 4) and B(–1, 6).
∴ PA = PB ...`[∵ "By distance formula, distance" = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)]`
⇒ (PA)2 = (PB)2
⇒ (– 5 – h)2 + (4 – k)2 = (– 1 – h)2 + (6 – k)2
⇒ 25 + h2 + 10h + 16 + k2 – 8k = 1 + h2 + 2h + 36 + k2 – 12k
⇒ 25 + 10h + 16 – 8k = 1 + 2h + 36 – 12k
⇒ 8h + 4k + 41 – 37 = 0
⇒ 8h + 4k + 4 = 0
⇒ 2h + k + 1 = 0 ...(i)
Mid-point of AB = `((-5 - 1)/2, (4 + 6)/2)` = (– 3, 5) ...`[∵ "Mid-point" = ((x_1 + x_2)/2, (y_1 + y_2)/2)]`
At point (– 3, 5), from equation (i),
2h + k = 2(– 3) + 5
= – 6 + 5
= – 1
⇒ 2h + k + 1 = 0
So, the mid-point of AB satisfy the equation (i).
Hence, infinite number of points, in fact all points which are solution of the equation 2h + k + 1 = 0, are equidistant from the points A and B.
Replacing h, k by x, y in above equation, we have 2x + y + 1 = 0
APPEARS IN
संबंधित प्रश्न
Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
Find the distance between the following pair of points:
(-6, 7) and (-1, -5)
Find the distance between the following pair of points:
(a+b, b+c) and (a-b, c-b)
If the point A(x,2) is equidistant form the points B(8,-2) and C(2,-2) , find the value of x. Also, find the value of x . Also, find the length of AB.
Using the distance formula, show that the given points are collinear:
(-2, 5), (0,1) and (2, -3)
Determine whether the point is collinear.
R(0, 3), D(2, 1), S(3, –1)
Find the distance of the following point from the origin :
(0 , 11)
Find the distance between the following point :
(sin θ , cos θ) and (cos θ , - sin θ)
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
P(5 , -8) , Q (2 , -9) and R(2 , 1) are the vertices of a triangle. Find tyhe circumcentre and the circumradius of the triangle.
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.
A point A is at a distance of `sqrt(10)` unit from the point (4, 3). Find the co-ordinates of point A, if its ordinate is twice its abscissa.
The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.
Give the relation that must exist between x and y so that (x, y) is equidistant from (6, -1) and (2, 3).
Find distance between point A(– 3, 4) and origin O
Find distance CD where C(– 3a, a), D(a, – 2a)
Show that A(1, 2), (1, 6), C(1 + 2 `sqrt(3)`, 4) are vertices of a equilateral triangle
The point P(–2, 4) lies on a circle of radius 6 and centre C(3, 5).
Show that Alia's house, Shagun's house and library for an isosceles right triangle.
