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प्रश्न
Prove that the points (a, b), (a1, b1) and (a −a1, b −b1) are collinear if ab1 = a1b.
उत्तर
The formula for the area ‘A’ encompassed by three points, `(x_1,y_1),(x_2,y_2),and (x_3,y_3)` is given by the formula,
`triangle=1/2[(x_1y_2+x_2y_3+x_3y_1)-(x_2y_1+x_3y_2+x_1y_3)]`
If three points are collinear the area encompassed by them is equal to 0.
The three given points are(a,b),(a-a1 b-b1)and(a-a1,b-b1) If they are collinear then the area enclosed by them should be 0.
`triangle =1/2[ab_1+a_1(b-b_1)+(a-a_1)b)-(a_1b+(a-a_)b_1+a(b-b_1))]`
`0=1/2[(ab_1+a_1b-a_1b_1+ab-a_1b)-(a_1b+ab1-a_-1b_1-ab-ab_1)]`
`0=1/2[ab_1+a_1b-a_1b_1+ab-a_1b-a_1b-a_1b+a_1b_1-ab+ab_1]`
`0=ab_1-a_1b`
`ab_1=a_1b`
Hence we have proved that for the given conditions to be satisfied we need to have
`a_1b=ab_1 `
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