• # question_answer The equation of $2{{x}^{2}}+3{{y}^{2}}-8x-18y+35=k$ represents [IIT 1994] A) No locus if $k>0$                                B) An ellipse, if $k<0$ C) A point if,$k=0$                                   D) A hyperbola, if $k>0$

Given equation, $2{{x}^{2}}+3{{y}^{2}}-8x-18y+35-k=0$ Compare with $a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0$,we get $a=2,\,\,\,b=3,\,\,\,h=0,\,\,\,g=-4,\,\,\,f=-9,\,\,\,c=35-k$ $\Delta =abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}$ $=6(35-k)+0-162-48-0$ $\Delta =210-6k-210=-6k$; $\Delta =0$, if $k=0$ So, that given equation is a point if $k=0$.