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If a ,b ,c are the p t h ,q t h ,r t h t...

If `a ,b ,c` are the `p t h ,q t h ,r t h` terms, respectively, of an `HP` , show that the points `(b c ,p),(c a ,q),` and `(a b ,r)` are collinear.

Text Solution

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Let the first term be A and the common difference be D of the corresponding AP.
Given pth term of `HP=a`
`therefore` pth term `AP=(1)/(a)`
`therefore A + (p-1)D=(1)/(a)`
Similarly, `A+(q-1)D=(1)/(b)`
`A+(r-1)D=(1)/(c)`
Subtracting,we get
`(p-q)D=(1)/(a)=(1)/(b)`
and `(q-r)D=(1)/(b)-(1)/(c)`
Dividing, we get
`(p-q)/(q-r) =(bc-ac)/(ac-ab)`
and Slope of `BC =(r-q)/(ab-ca)=(q-p)/(ca-bc)` [Using (1)]
`therefore` Slope of BC =Slope of AB
Therefore, A, B, and C are collinear.
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