Question

If the equation $x^2+y^2+2gx+2fy+1=0$ represent a pair of straight lines, then

• A. $g^2–f^2=1$
• B. $f^2–g^2=1$
• C. $g^2+f^2=1$
• D. $g^2+f^2=\frac{1}{2}$

Answer 1

The correct option is C

$g^2+f^2=1$

Explanation for correct option

The general equation of the pair of straight line is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of straight line if $\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0$

Given equation of a pair of straight lines is $x^2+y^2+2gx+2fy+1=0$

Now comparing the equation with the standard form we have $a=1,b=1,h=0,g=g,f=f,c=1$

$$\begin{gathered} \therefore \left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|=0 \\ \Rightarrow \left|\begin{array}{ccc}1& 0& g\\ 0& 1& f\\ g& f& 1\end{array}\right|=0 \\ \Rightarrow 1\left(1-f^2\right)+g\left(0-g\right)=0 \\ \Rightarrow 1-f^2-g^2=0 \\ \Rightarrow g^2+f^2=1 \end{gathered}$$

Hence, Option (C) is correct i.e. $g^2+f^2=1$