If `(a)/(b) = (c)/(d)`, show that : `(a + b) : (c + d) = sqrt(a^(2) + b^(2)) : sqrt(c^(2) + d^(2))`

Given four quantities a , b , c and d are in proportion. Show that : (a - c)b^(2) : (b - d)cd = (a^(2) - b^(2) - ab) : (c^(2) - d^(2) - cd)

(i) If a , b , c are in continued proportion, show that : (a^(2) + b^(2))/(b(a+c)) = (b(a + c))/(b^(2) + c^(2)) . (ii) If a , b , c are in continued proportion and a(b - c) = 2b , prove that : a - c = (2(a + b))/(a) . (iii) If (a)/(b) = (c)/(d) show that : (a^(3)c + ac^(3))/(b^(3)d +bd^(3)) = ((a + c)^(4))/((b + d)^(4)) .

if a , b , c and d are in proportion, prove that : (i) (a-b)/(c-d)= sqrt((3a^(2) + 8b^(2))/(3c^(2) + 8d^(2))) (ii) ((5a^(2) + 12c^(2))/(5b^(2) + 12d^(2))) = sqrt((3a^(4) - 7c^(4))/(3b^(4) - 7d^(4)))

(a) a^2+b^2 (b) a+b (c) a^2-b^2 (d) sqrt(a^2+b^2)

if a : b = e : d , show that : 3a + 2b : 3a - 2b = 3c + 2d : 3c - 2d .

If a, b, c ,d be in G.P. , show that (i) (b -c)^(2) + (c - a)^(2) +(d -b)^(2) = (a - d)^(2) (ii) a^(2) + b^(2) + c^(2) , ab + bc + cd , b^(2) + c^(2) + d^(2) are in G.P.

If a,b,c,d be in G.P. show that (b-c)^2 + (c-a)^2 + (d-b)^2 = (a-d)^2 .

If a, b, c, d are in G. P., show that they are also in G. P. a^(2)+b^(2),b^(2) +c^(2), c^(2)+d^(2)

If a ,\ b ,\ c ,\ d are in G.P., show that: (a b+b c+c d)^2=(a^2+b^2+c^2)(b^2+c^2+d^2)

The length of the perpendicular drawn from the point P(a , b , c) from z-axis is a. sqrt(a^2+b^2) b. sqrt(b^2+c^2) c. sqrt(a^2+c^2) d. sqrt(a^2+b^2+c^2)