If l my - nz - lx - m nz - lx - my - nlx - my - nz - prove y-z-x-l - z-x-y-m - x-y - z-n -

l(my+nz lx) = m(nz+lx my) = n(lx+my nz) Dividing by lmn , we get my+nz lxmn = nz+lx mynl = lx+my nzlm ⇒ #160; (my+nz lx) + (nz+lx my)mn+nl ...

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Byju's Answer

Standard VIII

Mathematics

Addition and Subtraction of Algebraic Expression

If l my + n...

Question

If $l (m y + n z - l x) = m (n z + l x - m y) = n (l x + m y - n z),$ prove $\frac{y + z - x}{l} = \frac{z + x - y}{m} = \frac{x + y - z}{n} .$

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Solution

$l (m y + n z - l x) = m (n z + l x - m y) = n (l x + m y - n z)$

Dividing by $l m n$ , we get

$\frac{m y + n z - l x}{m n} = \frac{n z + l x - m y}{n l} = \frac{l x + m y - n z}{l m}$

$\Rightarrow \frac{(m y + n z - l x) + (n z + l x - m y)}{m n + n l} = \frac{(n z + l x - m y) + (l x + m y - n z)}{n l + l m} = \frac{(l x + m y - n z) + (m y + n z - l x)}{l m + m n}$

$\Rightarrow \frac{2 n z}{n (m + l)} = \frac{2 l x}{l (n + m)} = \frac{2 m y}{m (l + n)}$

$\Rightarrow \frac{2 z}{m + l} = \frac{2 x}{n + m} = \frac{2 y}{l + n}$

Thus, adding and subtracting the three fractions, we have

$\frac{y + z - x}{l} = \frac{z + x - y}{m} = \frac{x + y - z}{n}$

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Similar questions

If l, m, n are the direction cosines of the normal to the plane and p be the perpendicular distance of the plane from the origin, then the equation of the plane is:

Q. Let P be any point on the plane lx + my + nz = p and Q be a point on the line OP such that

O P \cdot O Q = p^{2}

. The locus of the point Q is

Q. If

n > m, y > n, z > r, (x, y, z > 0)

such that

∣ ∣ ∣ ∣ \begin{matrix} x & n & r m & y & r m & n & z \end{matrix} ∣ ∣ ∣ ∣ = 0

then greatest value of

\frac{27 x y z}{(x - m) (y - n) (z - r)}

(1 + x)^{I} (1 + y)^{m} (1 + z)^{n} \geq \frac{1}{I x + m y + n z}

for I , m , n > 0 and

ϵ

N , x , y ,z > 0

Q. If

ξ = l x + m y + n z

η = n x + l y + m z

ζ = m x + n y + l z

, and if the same equations are true for all values of

x, y, z

when

ξ, η, ζ

are interchanged with

x, y, z

respectively, show that

l^{2} + 2 m n = 1

m^{2} + 2 l n = 1

n^{2} + 2 l m = 1

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If l(my+nz−lx)=m(nz+lx−my)=n(lx+my−nz), prove y+z−xl=z+x−ym=x+y−zn.

If $l (m y + n z - l x) = m (n z + l x - m y) = n (l x + m y - n z),$ prove $\frac{y + z - x}{l} = \frac{z + x - y}{m} = \frac{x + y - z}{n} .$