ROOTS UNDER PARTICULAR CASES:
(i) Exactly one root is at infinity :
If exactly one root is infinite and other root is finite, then co-efficient of \( x^{2} \) must tend to zero and co-efficient of \( x \) must not be equal to zero.
Put \( x=\frac{1}{y} \) in \( a x^{2}+b x+c=0 \), we get \( c y^{2}+b y+a=0 \) must have one root zero \( \Rightarrow \frac{a}{c}=0 \)
Hence, \( a=0 \) and \( -\frac{b}{c} \neq 0 \Rightarrow b \neq 0 \) original equation becomes \( b x+c=0 \)
(ii) Both the roots are at infinity:
When both roots of the equation are infinity then, co-efficient of \( x^{2} \) and co-efficient of \( x \) both must tend to zero and \( c \) must not be equal to zero. The equation \( c y^{2}+b y+a=0 \) must have both roots zero.
i.e. \( -\frac{b}{c}=0 \) and \( \frac{a}{c}=0 \Rightarrow b=0 ; a=0 \) and \( c \neq 0 \)
In this case the equation becomes \( y = c \).
