First slide

Equation of line in Straight lines

Question

$If the straight lines x + y - 2 = 0, 2 x - y + 1 = 0, and a x + b y - c = 0 are concurrent, then the family of lines$
$2 a x + 3 b y + c = 0 (a, b, c are nonzero) is concurrent at$

Moderate

A

(2,3)
B

(1/2,1/3)
C

(-1/6 ,- 5/9)
D

(2/3,-7/5)

Solution

$G i v e n t h a t t h r e e l i n e s a r e c o n c u r r e n t h e n c e,$
$\begin{array}{l} |\begin{array}{ccc} 1 & 1 & - 2 \\ 2 & - 1 & 1 \\ a & b & - c \end{array}| = 0 \\ E x p a n d t h e d e t e r m i n a n t, w e g e t \\ or a + 5 b - 3 c = 0 \\ or - \frac{a}{3} - \frac{5}{3} b + c = 0 - - - (1) \end{array}$
$2 a x + 3 b y + c = 0 - - - - (2) comparing (1) and (2)$
$h e n c e t h e s t r a i g h t l i n e s a r e concurrent at 2 x = - \frac{1}{3} and 3 y = - \frac{5}{3}$
$So, x = - \frac{1}{6}, y = - \frac{5}{9}$

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