Prove without expansion that `|a h+bgga b+c h bf+b afh b+b c af+b cc bg+fc|=a|a h+bga h bf+b a h b af+b cgf|`

Prove that a^4+b^4+c^4> a b c(a+b+c),w h e r ea ,b ,c > 0.

If a^2+b^2+c^2-a b-b c-c a=0,\ p rov e\ t h a t\ a=b=c

a ,b ,c x in R^+ such that a ,b ,a n dc are in A.P. and b ,ca n dd are in H.P., then a b=c d b. a c=b d c. b c=a d d. none of these

Through a point A on the x-axis, a straight line is drawn parallel to the y-axis so as to meet the pair of straight lines a x^2+2h x y+b y^2=0 at B and Cdot If A B=B C , then (a) h^2=4a b (b) 8h^2=9a b (c) 9h^2=8a b (d) 4h^2=a b

Through a point A (2,0) on the x-axis, a straight line is drawn parallel to the y-axis so as to meet the pair of straight lines a x^2+2h x y+b y^2=0 at B and Cdot If A B=B C , then (a) h^2=4a b (b) 8h^2=9a b (c) 9h^2=8a b (d) 4h^2=a b

If a ,b ,c in R^+, t h e n(b c)/(b+c)+(a c)/(a+c)+(a b)/(a+b) is always

Through a point A on the x-axis, a straight line is drawn parallel to the y-axis so as to meet the pair of straight lines a x^2+2h x y+b y^2=0 at B and Cdot If A B=B C , then h^2=4a b (b) 8h^2=9a b 9h^2=8a b (d) 4h^2=a b

(i) a , b, c are in H.P. , show that (b + a)/(b -a) + (b + c)/(b - c) = 2 (ii) If a^(2), b^(2), c^(2) are A.P. then b + c , c + a , a + b are in H.P. .

If a,b,c are in H.P. , then

If a, b, c are in AP amd b, c, a are in GP , then show that c, a, b are in H.P. Find a : b : c .