Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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# If the Lines X + Ay + a = 0, Bx + Y + B = 0 and Cx + Cy + 1 = 0 Are Concurrent, Then Write the Value of 2abc − Ab − Bc − Ca. - Mathematics

Answer in Brief

If the lines x + ay + a = 0, bx + y + b = 0 and cx + cy + 1 = 0 are concurrent, then write the value of 2abc − ab − bc − ca.

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#### Solution

The given lines are
x + ay + a = 0        ... (1)
bx + y + b = 0        ... (2)
cx + cy + 1 = 0       ... (3)
It is given that the lines (1), (2) and (3) are concurrent.

$\therefore \begin{vmatrix}1 & a & a \\ b & 1 & b \\ c & c & 1\end{vmatrix} = 0$

$\Rightarrow \left( 1 - bc \right) - a\left( b - bc \right) + a\left( bc - c \right) = 0$

$\Rightarrow 1 - bc - ab + abc + abc - ac = 0$

$\Rightarrow 2abc - ab - bc - ca = - 1$

Hence, the value of 2abc − ab − bc − ca is −1

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Q 7 | Page 132
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