Question

If $=\left|\begin{array}{ccc}\frac{1}{(a+x)}& \frac{1}{(b+x)}& \frac{1}{(c+x)}\\ \frac{1}{(a+y)}& \frac{1}{(b+y)}& \frac{1}{(c+y)}\\ \frac{1}{(a+z)}& \frac{1}{(b+z)}& \frac{1}{(c+z)}\end{array}\right|=\frac{P}{Q}$, where Q is the product of denominators. Find P

• A. $(b-c)(c-a)(a-b)(y+z)(z+x)(x+y)$
• B. $(b-c)(c-a)(a-b)(y-z)(z-x)(x-y)$
• C. $(b+c)(c+a)(a+b)(y-z)(z-x)(x-y)$
• D. $(b+c)(c+a)(a+b)(y+z)(z+x)(x+y)$

Answer 1

The correct option is B $(b-c)(c-a)(a-b)(y-z)(z-x)(x-y)$

$\Delta =\left|\begin{array}{ccc}\frac{1}{(a+x)}& \frac{1}{(b+x)}& \frac{1}{(c+x)}\\ \frac{1}{(a+y)}& \frac{1}{(b+y)}& \frac{1}{(c+y)}\\ \frac{1}{(a+z)}& \frac{1}{(b+z)}& \frac{1}{(c+z)}\end{array}\right|$

Apply $C_2\to C_2-C_1;C_3\to C_3-C_1$

$=\left|\begin{array}{ccc}\frac{1}{(a+x)}& \frac{(a-b)}{(a+x)(b+x)}& \frac{(a-c)}{(a+x)(c+x)}\\ \frac{1}{(a+y)}& \frac{(a-b)}{(a+y)(b+y)}& \frac{(a-c)}{(a+y)(c+y)}\\ \frac{1}{(a+z)}& \frac{(a-b)}{(a+z)(b+z)}& \frac{(a-c)}{(a+z)(c+z)}\end{array}\right|$

$=\left|\begin{array}{ccc}\frac{1}{(a+x)}& \frac{1}{(a+x)(b+x)}& \frac{1}{(a+x)(c+x)}\\ \frac{1}{(a+y)}& \frac{1}{(a+y)(b+y)}& \frac{1}{(a+y)(c+y)}\\ \frac{1}{(a+z)}& \frac{1}{(a+z)(b+z)}& \frac{1}{(a+z)(c+z)}\end{array}\right|$

$=\frac{(a-b)(a-c)}{Q}\left|\begin{array}{ccc}(b+x)(c+x)& c+x& b+x\\ (b+y)(c+y)& c+y& b+y\\ (b+z)(c+z)& c+z& b+z\end{array}\right|$

apply $R_2\to R_2-R_1;R_3\to R_3-R_1$

$=\frac{(a-b)(a-c)}{Q}\left|\begin{array}{ccc}(b+x)(c+x)& c+x& b+x\\ (y-x)(y+x+b+c)& (y-x)& (y-x)\\ (z-x)(z+x+b+c)& (z-x)& (z-x)\end{array}\right|$

$=\frac{(a-b)(a-c)(y-x)(z-x)}{Q}\left|\begin{array}{ccc}(b+x)(c+x)& c+x& b+x\\ y+x+b+c& 1& 1\\ z+x+b+c& 1& 1\end{array}\right|$

apply $R_3\to R_3-R_2$

$=\frac{(a-b)(a-c)(y-x)(z-x)}{Q}\left|\begin{array}{ccc}(b+x)(c+x)& c+x& b+x\\ y+x+b+c& 1& 1\\ (z-y)& 0& 0\end{array}\right|$

$=\frac{(a-b)(a-c)(y-x)(z-x)(z-y)}{Q}\left|\begin{array}{cc}c+x& b+x\\ 1& 1\end{array}\right|$

$=\frac{(a-b)(a-c)(y-x)(z-x)(z-y)(c-b)}{Q}$

$\Delta =\frac{(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)}{Q}=\frac{P}{Q}$ (given)

Hence $P=(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)$