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181. In what ratio does the point (-4, 6) divide the line segment joining the points A(-6, 10) and B(3, -8).

Let P(-4, 6) divide the line segment joining the points A(-6, 10) and B(3, -8) in the ratio K : 1.
Then, the co-ordinates of P are given by

$straight P open square brackets fraction numerator straight m subscript 1 straight x subscript 2 plus straight m subscript 2 straight x subscript 1 over denominator straight m subscript 1 plus straight m subscript 2 end fraction comma fraction numerator straight m subscript 1 straight y subscript 2 plus straight m subscript 2 straight y subscript 1 over denominator straight m subscript 1 plus straight m subscript 2 end fraction close square brackets$
Here, we have
x₁ = -6,    y₁ = 10
x₂ = 3,    y₂ = -8
and    m₁ = K    m₂ = 1
So, the co-ordinate of P are

$straight P open square brackets fraction numerator straight K left parenthesis 3 right parenthesis plus left parenthesis 1 right parenthesis left parenthesis negative 6 right parenthesis over denominator straight K plus 1 end fraction comma fraction numerator straight K cross times left parenthesis negative 8 right parenthesis plus left parenthesis 1 right parenthesis 10 over denominator straight K plus 1 end fraction close square brackets equals straight P open square brackets fraction numerator 3 straight K minus 6 over denominator straight K plus 1 end fraction comma space fraction numerator negative 8 straight K plus 10 over denominator straight K plus 1 end fraction close square brackets$

But the co-ordinates of P are given as (-4, 6)

$therefore space space fraction numerator 3 straight K minus 6 over denominator straight K plus 1 end fraction equals negative 4 rightwards double arrow space 3 straight K minus 6 equals negative 4 left parenthesis straight K plus 1 right parenthesis rightwards double arrow 3 straight K minus 6 equals negative 4 straight K minus 4 rightwards double arrow 3 straight K plus 4 straight K equals negative 4 plus 6 rightwards double arrow 7 straight K equals 2 rightwards double arrow space straight K equals 2 over 7 and space space space space fraction numerator negative 8 straight k plus 10 over denominator straight k plus 1 end fraction equals 6 rightwards double arrow space minus 8 straight k plus 10 equals 6 left parenthesis straight k plus 1 right parenthesis rightwards double arrow negative 8 straight k plus 10 equals 6 straight k plus 6 rightwards double arrow negative 8 straight k minus 6 straight k equals 6 minus 10 rightwards double arrow negative 14 straight k equals negative 4 rightwards double arrow space straight k equals 4 over 14 equals 2 over 7 rightwards double arrow space straight k equals 2 over 7 space in space each space case.$

$space space therefore$ The required ratio is 2 : 7

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182. Find the area of triangle whose vertices are (1, -1), (-4, 6) and (-3, -5).

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183. Find the area of the triangle formed by the points P(-1.5, 3), Q(6, -2) and R(-3, 4).

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184.

Find the area of a triangle whose vertices are : (3, 8), (-4, 2) and (5, 1).

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185. Find the area of a triangle formed by the points A(5, 2), B(4, 7) and C(7, -4).

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186. Find the area of the triangle formed by the points (0, 0), (6, 0) and (4, 3).

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187. One end of a diameter of a circle is at (2,3) and the centre is (-2, 5), what are the coordinates of the other end of his diameter?

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188. The centre of a circle is (2α - 1, 7) and it passes through the point (- 3, - 1). If the diameter of the circle is 20 units, then find the value(s) of α.

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