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

If `a ,b ,c`are the `p t h ,q t h ,r t h`terms, respectively, of an `H

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