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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.

Let the given points of the line segment be A(5, -6) and B(-1, -4).
Let the y-axis cut AB at the point P in the ratio K : 1.
Then, the co-ordinates of P are given by

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Here, we have x₁ = 5,    y₁ = -6
x₂ = -1,    y₂ = -4
and    m₁ = K    m₂ = 1
Therefore, the coordinates of ‘P’ are given by

$straight P space open square brackets fraction numerator straight K left parenthesis negative 1 right parenthesis plus 1 left parenthesis 5 right parenthesis over denominator straight K plus 1 end fraction comma space fraction numerator straight K left parenthesis negative 4 right parenthesis plus 1 left parenthesis negative 6 right parenthesis over denominator straight K plus 1 end fraction close square brackets equals straight P space open square brackets fraction numerator negative straight K plus 5 over denominator straight K plus 1 end fraction comma space fraction numerator negative 4 straight K minus 6 over denominator straight K plus 1 end fraction close square brackets$

But P lies on y-axis. So its abscissa is 0

$therefore space space space space fraction numerator negative straight K plus 5 over denominator straight K plus 1 end fraction equals 0 rightwards double arrow space space space minus straight K space plus space 5 space equals space 0 rightwards double arrow space space minus straight K equals 5 rightwards double arrow space straight K space equals space 5$

Putting K = 5, we get the co-ordinates of P as

$equals space straight P space open square brackets fraction numerator negative 5 plus 5 over denominator 5 plus 1 end fraction comma space fraction numerator negative 4 left parenthesis 5 right parenthesis minus 6 over denominator 5 plus 1 end fraction close square brackets equals space straight P space open square brackets 0 over 6 comma space fraction numerator negative 20 space minus 6 over denominator 6 end fraction close square brackets equals space straight P open square brackets 0 comma space fraction numerator negative 26 over denominator 6 end fraction close square brackets equals straight P space open square brackets 0 comma space fraction numerator negative 13 over denominator 3 end fraction close square brackets$

∴ The point of intersection of AB and the y-axis is $straight P space open square brackets 0 comma space fraction numerator negative 13 over denominator 3 end fraction close square brackets$

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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).

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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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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.