Question

If ${}^{\prime }{\theta }^{\prime }$ is the parameter, then the family of lines represented by
$(2cos\theta +3sin\theta )X+(3cos\theta -5sin\theta )$
y - $(5cos\theta -7sin\theta )$ = 0 are concurrent at the point

Answer 1

$cos\theta (2x+3y-5)+sin\theta (3x-5y+7)=0$

$\Rightarrow (2x+3y-5)+tan\theta (3x-5y+7)=0$

of from $L_1+\lambda L_2=0,$ intersection point be P

$3\times (2x+3y-5=0)$

$2\times (3x-5y+7=0)$

___________________

$19y-29=0\Rightarrow y=\frac{29}{19}$ $\Rightarrow x=\frac{5y-7}{3}=\frac{145-133}{57}=\frac{12}{57}$

$\Rightarrow P(-\frac{4}{19},\frac{29}{19})$