Visual SolutionPYQ 2024 · Apr 8 Shift 2Standard

Question

If $\alpha
eq a, \beta
eq b, \gamma
eq c

and

\begin{vmatrix} \alpha & b & c \\ a & \beta & c \\ a & b & \gamma \end{vmatrix} = 0

, then

\frac{a}{\alpha - a} + \frac{b}{\beta - b} + \frac{\gamma}{\gamma - c}$ is equal to:

(A)

3

(B)

0

(C)

1

(D)

2

Solution Path

Row operations and expansion yield

\dfrac{a}{\alpha - a} + \dfrac{b}{\beta - b} + \dfrac{\gamma}{\gamma - c} = 0

01Question Setup

1/4

Given determinant

= 0

with $\alpha
eq a, \beta
eq b, \gamma
eq c

. Find

\dfrac{a}{\alpha - a} + \dfrac{b}{\beta - b} + \dfrac{\gamma}{\gamma - c}$.

\text{Expression} = \;?

02Row Operations

2/4

Apply

R_1 \to R_1 - R_2

and

R_2 \to R_2 - R_3

to simplify the determinant.

Simplified determinant

03SimplifyKEY INSIGHT

3/4

Expand and divide by

(\alpha - a)(\beta - b)(\gamma - c)

to get the expression equals

0

\text{Expression} = 0

04Final Answer

4/4

\dfrac{a}{\alpha - a} + \dfrac{b}{\beta - b} + \dfrac{\gamma}{\gamma - c} = 0

\boxed{0}

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