Question

6. Find the separate equations of lines
represented by
x2 + 2 (cosec a) xy + y2 = 0​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{Pair of lines is}$

$\mathsf{x^2+2\,cosecA\,xy+y^2=0}$

$\underline{\textsf{To find:}}$

$\textsf{Separate equations of given pair of lines}$

$\underline{\textsf{Solution:}}$

$\textsf{We find out the separate eqations by using square completion method}$

$\mathsf{consider,}$

$\mathsf{x^2+2\,cosecA\,xy+y^2=0}$

$\mathsf{x^2+2\,cosecA\,xy=-y^2}$

$\mathsf{Adding\;bothsides\;by\;y^2cosec^2A}$

$\mathsf{x^2+2\,cosecA\,xy+y^2cosec^2A=-y^2+y^2cosec^2A}$

$\mathsf{(x+cosecA\,y)^2=y^2(cosec^2A-1)}$

$\mathsf{(x+cosecA\,y)^2=y^2cot^2A}$

$\mathsf{(x+cosecA\,y)^2=(y\,cotA)^2}$

$\mathsf{Taking\;square\;root\;on\;bothsides\;we\;get}$

$\mathsf{x+cosecA\,y=\pm(y\,cotA)}$

$\mathsf{x+cosecA\,y\pm(y\,cotA=0}$

$\mathsf{x+(cosecA\pm\,cotA)y=0}$

$\underline{\textsf{Answer:}}$

$\mathsf{The\;separate\;equations\;are}$

$\mathsf{x+(cosecA+cotA)y=0}$

$\mathsf{x+(cosecA-cotA)y=0}$

$\underline{\textsf{Find more:}}$

Prove that the equation 2x^2 + 3xy - 2y^2 + 3x + y + 1 represents a pair of perpendicular lines and find the lines.

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