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Coordinate Geometry

Short Answer Type

221. In what ratio does the like x - y - 2 = 0 divides the line segment joining (3, -1) and (8, 9) ?

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222. Find the ratio in which the point (x, -1) divides the line segment joining the points (-3, 5) and (2, -5). Also find the value of x

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223. Find the ratio in which the point (2, y) divides the line segment joining the points A(-2, 2) and B (3, 7). Also find the value of y.

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224. In what ratio does the point P(2, -5) divide the line segment joining A(-3, 5) and B (4, -9)?

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225. The mid-points of the sides of a triangle are (3, 4), (4, 6) and (5, 7). Find the coordinates of the vertices of the triangle.

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226. Determine the ratio in which the line 3x + 4y - 9 = 0 divides the line segment joining the points (1, 3) and (2, 7)

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227. A line segment joining the points P(3, 3) and Q(6, - 6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.

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228. Find the area of the quadrilateral w hose vertices are A(0, 0), B(6, 0), C(4, 3) and D(0, 3).

Sol. Join BD.We know, area of ∆ABD
Here, we have x1 = 0, y1 = 0x2

Sol. Join BD.
We know, area of ∆ABD

$equals 1 half left square bracket straight x subscript 1 left parenthesis straight y subscript 2 minus straight y subscript 3 right parenthesis plus straight x subscript 2 left parenthesis straight y subscript 3 minus straight y subscript 1 right parenthesis plus straight x subscript 3 left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis right square bracket$

Here, we have x₁ = 0, y₁ = 0
x₂ = 6, y₂ = 0
and x₃ = 0, y₃ = 3
Now, area of ∆ABD

$equals 1 half left square bracket 0 left parenthesis 0 minus 3 right parenthesis plus 6 left parenthesis 3 minus 0 right parenthesis plus 0 left parenthesis 0 minus 0 right parenthesis right square bracket equals 1 half left square bracket 0 space straight x space minus 3 space plus space 6 space straight x space 3 space plus space 0 right square bracket equals 1 half left square bracket 0 plus 18 plus 0 right square bracket equals 1 half space straight x space 18 space minus 9 space sq. space units.$

We know that, area of ∆BCD
$equals 1 half left square bracket straight x subscript 1 left parenthesis straight y subscript 2 minus straight y subscript 3 right parenthesis plus straight x subscript 2 left parenthesis straight y subscript 3 minus straight y subscript 1 right parenthesis plus straight x subscript 3 left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis right square bracket$

Here, we have x₁ = 6,    y₁ = 0
x₂ = 4,    y₂ = 3
and    x₃ = 0,    y₃ = 3
Now. area of ∆BCD

$equals 1 half left square bracket 6 left parenthesis 3 minus 3 right parenthesis plus 4 left parenthesis 3 minus 0 right parenthesis plus 0 left parenthesis negative 3 right parenthesis right square bracket equals 1 half left square bracket 6 space straight x space 0 space plus space straight x space 3 space plus 0 space right square bracket equals 1 half left square bracket 0 plus 12 plus 0 right square bracket space equals 1 half space straight x space 12 space equals space 6 space sq. space units.$

Hence, the area of quadrilateral ABCD
= area (∆ABD) + area (∆BCD)
= (9 + 6) sq. units = 15 sq. units.

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229. Find the area of the quadrilateral whose vertices, taken in order are (-4, -2), (-3, -5), (3, -2) and (2, 3).

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230. Find the value of ‘K’ if the points (K, 3), (6, -2) and (-3, 4) are collinear.

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