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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).

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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.

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

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

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

Let the given points be A(-3, -4), B(3, 0) and C(-5, 0).

Now comma space space space AB space equals space square root of left parenthesis 3 plus 3 right parenthesis squared plus left parenthesis 0 plus 4 right parenthesis squared end root space space space space space space space space space space space space space space space space equals space square root of left parenthesis 6 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space space space space space space space space space space space space space space space space equals space square root of 36 plus 16 end root equals square root of 52 equals 2 square root of 13 space units BC thin space equals space square root of left parenthesis negative 5 minus 3 right parenthesis squared plus 0 end root equals square root of left parenthesis negative 8 right parenthesis squared end root equals square root of 64 equals 8 space units and space CA space equals space square root of left parenthesis negative 3 plus 5 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root equals square root of left parenthesis 2 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root space space space space space space space space space space space equals space square root of 4 plus 16 end root equals square root of 20 equals 2 square root of 5 space units

Since, AB ≠ BC ≠ CA
Therefore, given points are the vertices of scalene triangle.
Now,

$Area space of space increment ABC space equals space space space 1 half space straight x space 8 space straight x 4 equals space 4 space straight x space 4 space equals space 16 space sq. space units.$

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

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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).

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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

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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.

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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.

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