Question

find the equation of the parabola with focus is( 2 ,3 )and directrix is x - 2y - 6 =0

Answer 1

(x+1)2+(y+2)2=[x−2y+312+22−−−−−√]2(x+1)2+(y+2)2=[x−2y+312+22]2

5[(x2+2x+1)+(y2+2y+4)]=(x−2y+3)25[(x2+2x+1)+(y2+2y+4)]=(x−2y+3)2

⇒5[(x2+2x+1)+(y2+4y+4)]=x2+4y2+9−4xy−12y+6x⇒5[(x2+2x+1)+(y2+4y+4)]=x2+4y2+9−4xy−12y+6x

⇒5x2+10x+5+5y2+20y+20=x2+4y2+9−4xy−12y+6x⇒5x2+10x+5+5y2+20y+20=x2+4y2+9−4xy−12y+6x

4x2+y2+4xy+4x+32y+16=04x2+y2+4xy+4x+32y+16=0