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Coordinate Geometry

Short Answer Type

211. Find (he ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also find the point of intersection.

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212. Find the ratio in which the line segment joining the points (6, 4) and (1, - 7) is divided by x-axis. Also find the point of intersection.

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213. If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram taken in order, find the value of p.

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214. Prove (hat the points (-2, -1), (1, 0), (4, 3) and (1, 2) are the vertices of a parallelogram is it a rectangle?

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215. The three vertices of a rhombus, taken in order are (2, -1), (3, 4) and (-2, 3). Find the fourth vertex, is it a square?

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216. The mid-points of the sides of a triangle are (3, 4), (4, 1) and (2, 0). Find the coordinates of the vertices of the triangle.

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217. Find the lengths of the medians of the triangle whose vertices are (1, -1), (0, 4) and (-5, 3).

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218. Determine the ratio in which the line 3x + y - 9 = 0 divides the segment joining the points (1, 3) and (2, 7).

Let the given points be A( 1, 3) and B(2, 7).

Let the line 3x + y - 9 = 0 divides the line segment AB in the ratio K : 1 at point P. Then the co-ordinates of P are given by

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Here, we have
x₁ = 1,    y₁ = 3
x₂ = 2,    y₂ = 7
and    m₁ = K    m₂ = 1
So, co-ordinates of P are

$straight P space open square brackets fraction numerator straight K left parenthesis 2 right parenthesis plus 1 left parenthesis 1 right parenthesis over denominator straight K plus 1 end fraction comma space fraction numerator straight K left parenthesis 7 right parenthesis plus 1 left parenthesis 3 right parenthesis over denominator straight K plus 1 end fraction close square brackets equals space straight P space open square brackets fraction numerator 2 straight K plus 1 over denominator straight K plus 1 end fraction comma space fraction numerator 7 straight K plus 3 over denominator straight K plus 1 end fraction close square brackets$

But P lies on 3x + y - 9 = 0. ...(i)
Substituting the values of x (abscissa) and y (ordinate) in (i), we get

$3 space open parentheses fraction numerator 2 straight K plus 1 over denominator straight K plus 1 end fraction close parentheses plus open parentheses fraction numerator 7 straight K plus 3 over denominator straight K plus 1 end fraction close parentheses minus 9 space space equals space 0 rightwards double arrow space space space fraction numerator 3 left parenthesis 2 straight K plus 1 right parenthesis plus left parenthesis 7 straight K plus 3 right parenthesis over denominator straight K plus 1 end fraction minus 9 equals 0 rightwards double arrow space fraction numerator left parenthesis 6 straight K plus 3 plus 7 straight K plus 3 right parenthesis minus 9 left parenthesis straight K plus 1 right parenthesis over denominator straight K plus 1 end fraction equals 0 rightwards double arrow space 6 straight K plus 7 straight K plus 6 minus 9 straight K minus 9 equals 0 rightwards double arrow space space space space space space space 4 straight K minus 3 equals 0 rightwards double arrow space space space space space space space space space space 4 straight K space equals space 3 rightwards double arrow space space space space space space straight K equals 3 over 4$
So, the required ratio be (3 : 4).

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219. 26. If C is a point lying on the line segment AB joining A(1, 1) and B(2, - 3) seen that 3AC = CB, then find the co-ordinates of C.

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220. Prove that the mid-point of the hypotenuse of a right angled triangle is equidistant from its vertices.

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