Question

If `a/b = c/d` then prove that each of the given ratio is equal to `sqrt((4a^2 - 3c^2)/(4b^2 - 3d^2))`.

Answer 1

Let `a/b = c/d` = k

Then,

a = bk

c = dk

`sqrt((4a^2 - 3c^2)/(4b^2 - 3d^2)) = sqrt((4(bk)^2 - 3(dk)^2)/(4b^2 - 3d^2)`

= `sqrt((4b^2k^2 - 3d^2k^2)/(4b^2 - 3d^2))`

= `sqrt((k^2(4b^2 - 3d^2))/(4b^2 - 3d^2))`

= `sqrt(k^2)`

= k

Since we established that the original ratios `a/b and c/d` both equal k, and the new ratio `sqrt((4a^2 - 3c^2)/(4b^2 - 3d^2))` also equals k they are all equal to each other:

`a/b = c/d = sqrt((4a^2 - 3c^2)/(4b^2 - 3d^2))`