If y is the mean proportional between x and z- prove that - x2-y2-z2-x-2-y-2-z-2 - y4

Given. y is mean proportion between x and z so y^2=xz we have to prove that (x^2−y^2+z^2)/(x^−2−y^−2+z^−2) = y^4 Taking L. H. S we have → (x^2−y^2+z^2)/(x^− ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard X

Mathematics

Proportion

If y is the m...

Question

If y is the mean proportional between x and z, prove that : $\frac{x^{2} - y^{2} + z^{2}}{x^{- 2} - y^{- 2} + z^{- 2}} = y^{4}$

Open in App

Solution

Given.
y is mean proportion between x and z
so
$y^{2} = x z$
we have to prove that
$\frac{(x^{2} - y^{2} + z^{2})}{(x^{-} 2 - y^{-} 2 + z^{-} 2)} = y^{4}$
Taking L. H. S we have
$\to \frac{(x^{2} - y^{2} + z^{2})}{(x^{-} 2 - y^{-} 2 + z^{-} 2)}$

$\to \frac{x^{2} + y^{2} + z^{2}}{\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}}}$
$\to \frac{x^{2} - x z + z^{2}}{\frac{1}{x^{2}} - \frac{1}{x z} + \frac{1}{z^{2}}}$

$\to \frac{x^{2} - x z + z^{2}}{z^{2} - x z + z^{2}}$

$\to x^{2} z^{2}$

$\to (x z)^{2}$

$\to (y^{2})^{2}$

$\to y^{4} = R . H . S$

Suggest Corrections

Similar questions

If y is the mean proportional between x and z ; show that xy + yz is the mean proportional between $x^{2} + y^{2}$ and $y^{2} + z^{2}$ .

Q. If x, y, z are in continued proportion, prove that :

\frac{(x + y)^{2}}{(y + z)^{2}} = \frac{x}{z}

Q. If

\frac{a}{y + z} = \frac{b}{z + x} = \frac{c}{x + y}

then prove that.

\frac{a (b - c)}{y^{2} - z^{2}} = \frac{b (c - a)}{z^{2} - x^{2}} = \frac{c (a - b)}{x^{2} - y^{2}}

Q. If

x y + y z + z x = 1

, the prove that

\frac{x}{1 + x^{2}} + \frac{y}{1 + y^{2}} + \frac{z}{1 + z^{2}} = \frac{2}{[(1 = x^{2}) (1 + y^{2}) (1 + z^{2})]^{1 / 2}}

Q. Show that

x^{2} + y^{2} + z^{2} - x y - y z - z x = \frac{1}{2} [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}]

Join BYJU'S Learning Program

If y is the mean proportional between x and z, prove that : x2−y2+z2x−2−y−2+z−2=y4

Given. y is mean proportion between x and z so y2=xz we have to prove that (x2−y2+z2)(x−2−y−2+z−2)=y4 Taking L. H. S we have →(x2−y2+z2)(x−2−y−2+z−2) →x2+y2+z21x2+1y2+1z2 →x2−xz+z21x2−1xz+1z2 →x2−xz+z2z2−xz+z2 →x2z2 →(xz)2 →(y2)2 →y4=R.H.S

If y is the mean proportional between x and z, prove that : $\frac{x^{2} - y^{2} + z^{2}}{x^{- 2} - y^{- 2} + z^{- 2}} = y^{4}$