Question

If a + b + c = 0, then the family of lines 3ax + by + 2c = 0 pass through fixed point

विकल्प

•  (2, 2/3)

• (2/3, 2)

•  (−2, 2/3)

• none of these

Answer 1

(2/3, 2)

Given:
a + b + c = 0
Substituting c = − a − b in 3ax + by + 2c = 0, we get:

\[3ax + by - 2a - 2b = 0\]

\[ \Rightarrow a\left( 3x - 2 \right) + b\left( y - 2 \right) = 0\]

\[ \Rightarrow \left( 3x - 2 \right) + \frac{b}{a}\left( y - 2 \right) = 0\]

This line is of the form

 \[L_1 + \lambda L_2 = 0\],  which passes through the intersection of the lines \[L_1 \text { and } L_2\] i.e. 

\[3x - 2 = 0 \text { and } y - 2 = 0\].

Solving \[3x - 2 = 0 \text { and } y - 2 = 0\],we get:

\[x = \frac{2}{3}, y = 2\]

Hence, the required fixed point is \[\left( \frac{2}{3}, 2 \right)\].