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  • Need Solution for R. D. Sharma Maths Class 12 Chapter determinants Exercise 5.2 Question 54 Maths Textbook Solution.

Need Solution for R. D.  Sharma Maths Class 12 Chapter determinants  Exercise 5.2 Question 54 Maths Textbook Solution.

Answers (1)

Answer: 2

Hint: Use determinant formula

Given:\left|\begin{array}{ccc} a & b-y & c-z \\ a-x & b & c-z \\ a-x & b-y & c \end{array}\right|=0 \text { then using property of determinant. Find } \frac{a}{x}+\frac{b}{y}+\frac{c}{z} \text { where } x, y, z=0

Solution:

\begin{aligned} &\text { Take } x, y, z \text { common from } C_{1}, C_{2}, C_{3}\\ &(x y z)\left|\begin{array}{ccc} \frac{a}{x} & \frac{b}{y}-1 & \frac{c}{z}-1 \\ a-\frac{1}{x} & \frac{b}{y} & \frac{c}{z}-1 \\ \frac{a}{x}-1 & \frac{b}{y}-1 & \frac{c}{z} \end{array}\right|=0 \end{aligned}

\begin{aligned} &\text { Apply } \mathrm{C}_{1} \rightarrow C_{1}+C_{2}+C_{3} \\ &\mid \frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2 & \frac{b}{y}-1 & \frac{c}{z}-1 \\ &\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2 & \frac{b}{y} & \frac{c}{z}-1 \\ &\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2 & \frac{b}{y}-1 & \frac{c}{z} \end{aligned} \mid=0

\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2\right)\left|\begin{array}{ccc} 1 & \frac{b}{y}-1 & \frac{c}{z}-1 \\ 1 & \frac{b}{y} & \frac{c}{z}-1 \\ 1 & \frac{b}{y}-1 & \frac{c}{z} \end{array}\right|=0

\begin{aligned} &\text { Apply } \mathrm{R}_{2} \rightarrow R_{2}-R_{1} \& R_{3} \rightarrow R_{3}-R_{1} \\ &\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2\right)\left|\begin{array}{ccc} 1 & \frac{b}{y}-1 & \frac{c}{z}-1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right|=0 \end{aligned}

\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-2\right) \cdot 1=0 \text { or } \frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2

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