Question

Prove that the family of lines represented by
$x(1+\lambda )+y(2-\lambda )+5=0$,
$\lambda$ being arbitrary, pass through a fixed point. Also find the fixed point.

• A. $(\frac{5}{3},\frac{5}{3})$
• B. $(\frac{-5}{3},\frac{-5}{3})$
• C. $(\frac{-3}{5},\frac{-3}{5})$
• D. None of these

Answer 1

The correct option is B $(\frac{-5}{3},\frac{-5}{3})$

Given faxity of lines is

$x(1+\lambda )+y(2-\lambda )+5=0$

$x+\lambda x+2y-2y\lambda +5=0$

$(x+2y+5)+\lambda (x-y)=0$

we must have $x+2y+5=0$

$x=y$

so $3x+5=0x=-5/3,y=-5/3$

so it passes through $(-5/3,-5/3)$