What will be value of f(x) if \(\begin{vmatrix}x & b & c\\a & y & c\\a & b & z \end {vmatrix}\)?

Given, \(\begin{vmatrix}x & b & c\\a & y & c\\a & b & z \end {vmatrix}\) = \(\begin{vmatrix}x & b & c\\a – x & y – b & 0\\0 & b – y & z – c \end {vmatrix}\)

Applying the operation R₂ = R₂ – R₁ and R₃ = R₃ ? R₂
= (x ? a)(y ? b)(z ? c)\(\begin{vmatrix}x/(x – a) & b/(y – b) & c/(z – c)\\-1 & y 1 & 0\\0 & -1 & 1 \end {vmatrix}\)
Now, expanding the determinant we get,
= (x ? a)(y ? b)(z ? c)(\(\frac{x}{x-a} + \frac{b}{y – b} + \frac{c}{z-c}\))
= (x ? a)(y ? b)(z ? c)(\(\frac{x}{x-a} + \frac{b}{y – b} + \frac{c}{z-c}\) – 2)
This is because,
\(\frac{b}{y – b} + \frac{c}{z-c} = \frac{y-(y-b)}{y-b} + \frac{z-(z-c)}{z-c} = \frac{y}{y-b}\) – 1 + \(\frac{z}{z-c}\) – 1 = \(\frac{y}{y-b} + \frac{z}{z-c}\) – 2