Coordinate Geometry

Mathematics

Mathematics

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Find the distance between the points A and B in the following :

(i) A(2, 3), B(4, 1) (ii) A (a, b), B(-a, -b) (iii) A(5, -8); B(-7, -3)

(iv) A(4, 10), B(7, -6) (v) A (a + b, a - b), B(a - b, -a - b).

Solution not provided.

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Show that the points A(5, 6), B(1, 5), C(2, 1) and D(6, 2) arc the vertices of a square.

Hence, diagonal

AC = BD =

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Show that (1, -1) is the centre of the circle circumscribing the triangle whose angular points are (4, 3), (-2, 3) and (6, -1).

Let the given points be P(4, 3), Q(-2, 3) and R(6, -1). Let 0(1, -1) be the centre of the circle.

Fig. 7.25.

Here, we have

OP = OQ = OR

⇒ O is equidistant from P, Q and R.

Hence ‘O’ is the centre of the circle circumscribing the triangle.

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The line joining the points (1, -2) and (-3, 4) is trisected. Find the co-ordinates of the points of trisection.

Case I.

Fig. 7.28A.

Let the given points be A(3, -1) and B(-6, 5).

Let P and Q be the points of trisection of AB.

Then, AP = PQ = QB = 1

Thus ‘P’ divides AB in the ratio 1 : 2.

Here, we have x_{1} = 1, y_{1} = -2

x_{2} = -3, y_{2} = 4

and m_{1} = 1 m_{2} = 2

∴ The co-ordinates of ‘P’ are given by

Case II.

Fig. 7.29.

Now ‘Q’ divides AB in the ratio 2 : 1.

Here, we have x_{1} = 1, y = -2

x_{2} = -3, y_{2} = 4

and m_{1} = 2 m_{2} = 1

∴ The co-ordinates of ‘Q’ are given by

Hence, the co-ordinates of the points of trisection are

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Name the type of triangle formed, if any, by the following points and give reason for your answer:

Let the given points be

Since, AB = BC

So, the given points form an isosceles triangle.

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