Advertisements
Advertisements
प्रश्न
Using distance formula decide whether the points (4, 3), (5, 1) and (1, 9) are collinear or not.
Advertisements
उत्तर
Let A(x1, y1) = A(4, 3), B(x2, y2) = B(5, 1), C(x3, y3) = C(1, 9)
∴ d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt((5 -4)^2 + (1 - 3)^2`
= `sqrt(1^2 + (-2)^2`
= `sqrt(1+ 4)`
= `sqrt(5)` ...(i)
∴ d(B, C) = `sqrt((x_3 - x_2)^2 + (y_3 - y_2)^2`
= `sqrt((1 - 5)^2 + (9 - 1)^2`
= `sqrt((-4)^2 + 8^2`
= `sqrt(16 + 64)`
=`sqrt(80)`
= `4sqrt(5)` ...(ii)
∴ d(A, C) = `sqrt((x_3 -x_1)^2 + (y_3 - y_2)^2`
= `sqrt((1 - 4)^2 + (9 - 3)^2`
= `sqrt((-3)^2 + 6^2`
= `sqrt(9 + 36)`
= `sqrt(45)`
= `3sqrt(5)` ...(iii)
`sqrt(5) + 3sqrt(5) = 4sqrt(5)`
∴ d(A, B) + d(A, C) = d(B, C) ...[From (i), (ii) and (iii)]
∴ Points A, B, C are collinear.
APPEARS IN
संबंधित प्रश्न
If P (2, – 1), Q(3, 4), R(–2, 3) and S(–3, –2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus
Check whether (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.
Find the distance of a point P(x, y) from the origin.
The length of a line segment is of 10 units and the coordinates of one end-point are (2, -3). If the abscissa of the other end is 10, find the ordinate of the other end.
Prove that the points A(1, 7), B(4, 2), C(−1, −1), and D(−4, 4) are the vertices of a square.
A(–8, 0), B(0, 16) and C(0, 0) are the vertices of a triangle ABC. Point P lies on AB and Q lies on AC such that AP : PB = 3 : 5 and AQ : QC = 3 : 5. Show that : PQ = `3/8` BC.
Find the distance between the points:
A(1, –3) and B(4, –6)
Find the distance between the points `A((-8)/5, 2)` and `B(2/5, 2)`.
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
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 (1 ,1),(-4 , 4) and (4 , 6) are the certices of an isosceles triangle.
Prove that the points (0,3) , (4,3) and `(2, 3+2sqrt 3)` are the vertices of an equilateral triangle.
Find the distance between the points (a, b) and (−a, −b).
Find the distance between the origin and the point:
(8, −15)
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
The distance between points P(–1, 1) and Q(5, –7) is ______.
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`
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 equation of the perpendicular bisector of line segment joining points A(4,5) and B(-2,3) is ______.
