Question

If $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$, prove that $$\frac{x^2 + y^2 + z^2}{a^2 + b^2 + c^2} = \left(\frac{px + qy + rz}{pa + qb + rc}\right)^2$$.

Answer 1

Given: $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$$

To prove: $$\frac{x^2 + y^2 + z^2}{a^2 + b^2 + c^2} = \left(\frac{px + qy + rz}{pa + qb + rc}\right)^2$$

Proof:

1. Let $$\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = k$$, then $$x = ka, y = kb, z = kc$$
2. $$\text{L.H.S.} = \frac{x^2 + y^2 + z^2}{a^2 + b^2 + c^2} = \frac{k^2 a^2 + k^2 b^2 + k^2 c^2}{a^2 + b^2 + c^2} = \frac{k^2 (a^2 + b^2 + c^2)}{a^2 + b^2 + c^2} = k^2$$
3. $$\text{R.H.S.} = \left(\frac{px + qy + rz}{pa + qb + rc}\right)^2 = \left(\frac{p(ka) + q(kb) + r(kc)}{pa + qb + rc}\right)^2$$
4. $$\text{R.H.S.} = \left(\frac{k(pa + qb + rc)}{pa + qb + rc}\right)^2 = k^2$$
5. $$\text{L.H.S.} = \text{R.H.S.}$$

Hence proved.