Question

Write the condition for the quadratic equation which as roots at infinity​

Answer 1

Step-by-step explanation:

$\huge\red{\underline{\underline{\bf Correct \: question:-}}}$

Write the condition for the quadratic equation which has both roots at infinity.

$\huge\red{\underline{\underline{\bf Answer:-}}}$

$let, \\ \\ the \: \: following \: \: quadratic \: \: equation \: \: be \implies \\ \\ a {x}^{2} + bx + c \huge\{_ { \beta = \infty }^{ \alpha = \infty } \\ \\ let \: \: x = \frac{1}{y} \\ \\ \implies a {( \frac{1}{y} )}^{2} + b( \frac{1}{y} ) + c = 0 \\ \\ \implies c {y}^{2} + by + a = 0 \huge\{_ { \frac{1}{ \beta } = \delta = \frac{1}{ \infty } = 0 }^{ \frac{1}{ \alpha} = \gamma = \frac{1}{ \infty } = 0 } \\ \\ \leadsto \gamma + \delta = 0 \\ \\ \frac{ - b}{c} = 0 \\ \\ c \neq 0 \\ \\ \huge \boxed{ \implies b = 0} \\ \\ Also, \\ \\ \leadsto (\gamma )( \delta) = 0 \\ \\ \frac{a}{c} = 0 \\ \\ c \neq 0 \\ \\ \huge \boxed{ \implies a = 0} \\ \\ So, \: \: required \: \: condition \: \: is \implies \\ \\ \huge\boxed{a = 0, \: \: c \neq 0 \: \: and \: \: b = 0}$