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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.

Fig. 7.31.
Let the given points of the line segment be A(6, 4) and B(1, -7).
Let the x-axis cut AB at the point ‘P’ in the ratio K : 1.
Then the co-ordinates of ‘P’ are given as

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Here, we have x₁ = 6, y_x = 4
x₂ = 1, y₂ = -7
and m₁ = K. m₂ = 1
So, coordinate of P are

$open square brackets fraction numerator straight k left parenthesis 1 right parenthesis plus 1 left parenthesis 6 right parenthesis over denominator straight k plus 1 end fraction comma space fraction numerator straight k left parenthesis negative 7 right parenthesis plus 1 left parenthesis 4 right parenthesis over denominator straight k plus 1 end fraction close square brackets equals open square brackets fraction numerator straight K plus 6 over denominator straight K plus 1 end fraction comma space fraction numerator straight K left parenthesis negative 7 right parenthesis plus 1 left parenthesis 4 right parenthesis over denominator straight K plus 1 end fraction close square brackets$

But ‘P’ lies on x-axis. So its ordinate is 0.

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Putting $straight K equals 4 over 7 comma space$ we get the co-ordinates of 'P' as

$equals straight P space open square brackets fraction numerator begin display style 4 over 7 plus 6 end style over denominator begin display style 4 over 7 plus 1 end style end fraction comma space fraction numerator negative 7 open parentheses begin display style 4 over 7 end style close parentheses plus 4 over denominator begin display style 4 over 7 plus 1 end style end fraction close square brackets equals straight P space open square brackets fraction numerator begin display style fraction numerator 4 plus 42 over denominator 7 end fraction end style over denominator begin display style fraction numerator 4 plus 7 over denominator 7 end fraction end style end fraction comma space fraction numerator negative 4 plus 4 over denominator begin display style fraction numerator 4 plus 7 over denominator 7 end fraction end style end fraction close square brackets equals straight P space open square brackets 46 over 11 comma space 0 close square brackets$

∴ The point of intersection of AB and the x-axis be $open parentheses 46 over 11 comma space 0 close parentheses$

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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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