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Coordinate Geometry

Long Answer Type

201. 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.

Then comma space space space OP space equals space square root of left parenthesis 4 minus 1 right parenthesis squared plus left curly bracket 3 minus left parenthesis negative 1 right parenthesis right curly bracket squared end root rightwards double arrow space space space space space space space OP space equals space square root of left parenthesis 4 minus 1 right parenthesis squared plus left parenthesis 3 plus 1 right parenthesis squared end root rightwards double arrow space space space space space space space OP space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root rightwards double arrow space space space space space space space OP space equals space square root of 9 plus 16 end root equals square root of 25 equals 5 end root space space space space space space space space space space QQ space equals space square root of left parenthesis negative 2 minus 1 right parenthesis squared plus left parenthesis 3 plus 1 right parenthesis squared end root rightwards double arrow space space space space space space space QQ space equals square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root rightwards double arrow space space space space space space space QQ equals square root of 9 plus 16 end root equals square root of 25 equals 5 space and space space space space space space OR space equals space square root of left parenthesis 6 minus 1 right parenthesis squared plus left curly bracket negative 1 minus left parenthesis negative 1 right parenthesis right curly bracket squared end root space space rightwards double arrow space space space space space space space OR space space equals space square root of left parenthesis 5 right parenthesis squared plus left parenthesis negative 1 plus 1 right parenthesis squared end root rightwards double arrow space space space space space space space OR space space equals space square root of 25 plus 0 end root equals square root of 25 equals 5

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.

763 Views

Short Answer Type

202. If A(2, -1), B(3, 4), C(-2, 3) and D(-3, -2) be four points in a plane, show that ABCD is a rhombus but not a square.

228 Views

203. Show that the points (7, 10), (-2, 5) and (3, -4) are the vertices of an isosceles right triangle.

1273 Views

204. Show that A(-3, 2), B(-5, -5), C (2,-3) and D(4, 4) arc the vertices of a rhombus.

542 Views

205. Observe the graph given below and state whether triangle ABC is scalene, isosceles or equilateral. Justify your answer. Also find its area.

474 Views

206. Find the values of x for which the distance between the point P(2, -3) and Q(x, 5) is 10 units.

337 Views

207. Find a relation between x and y such thai the point P(x, y) is equidistant from the points A(2, 5) and B(-3, 7).

283 Views

208. If the distances of P (x, y) from the points A(3, 6) and B(-3, 4) are equal prove that 3x + y = 5

154 Views

209. If the points A (4, 3) and B (x, 5) are on the circle with the centre O (2,3), find the value of x.

Fig. 7.28.

978 Views

Long Answer Type

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

454 Views