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Question Number 46924 by 786786AM last updated on 02/Nov/18

If ((a−x)/(px)) = ((a−y)/(qy)) = ((a−z)/(rz)) and p, q, r are in AP, then x, y, z will be in

Ifa−xpx=a−yqy=a−zrzandp,q,rareinAP,thenx,y,zwillbein

Answered by tanmay.chaudhury50@gmail.com last updated on 02/Nov/18

((a−x)/(px))=((a−y)/(qy))=((a−z)/(rz))=(1/k)(say) px=k(a−x) p=((k(a−x))/x) q==((k(a−y))/y) r=((k(a−z))/z) 2q=p+r ((2k(a−y))/y)=((k(a−x))/x)+((k(a−z))/z) ((2(a−y))/y)=((a−x)/x)+((a−z)/z) ((2a)/y)−2=(a/x)−1+(a/z)−1 (2/y)=(1/x)+(1/z) so x,y,z are in HP

a−xpx=a−yqy=a−zrz=1k(say)px=k(a−x)p=k(a−x)xq==k(a−y)yr=k(a−z)z2q=p+r2k(a−y)y=k(a−x)x+k(a−z)z2(a−y)y=a−xx+a−zz2ay−2=ax−1+az−12y=1x+1zsox,y,zareinHP

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