Prove that the tangents to a circle x2 +y2+6y+4=0 at the point pf intersection with the line 3x+y=2 are perpendicular - Maths - Straight Lines

Question

Prove that the tangents to a circle x2 +y2+6y+4=0 at the point pf
intersection with the line 3x+y=2 are perpendicular to each other

Ghanshyam Dhakar · Accepted Answer

Dear student,
circle:-  x2+y2+6y+4=0    
1differentiate 2x+2ydydx+6dydx=0dydx = -xy+3  
2line:-  3x+y=2for intersection point put y=2-3x in eq 1  x2+2-3x2+62-3x+4=0x2+9x2+4-12x+12-18x+4=010x2-30x+20=0x2-3x+2=0x2-2x-x+2=0xx-2-1x-2=0x-1x-2=0x=1,2then use y=2-3x and eq2 x=1, y=2-3=-1, dydx=m1 = -xy+3=-1-1+3=-12x=2, y=2-6=-4, dydx=m2 = -xy+3=-2-4+3=2for perpendiculare m1×m2=-1LHS  =m1×m2=-12×2=-1=RHS proved 
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Prove that the tangents to a circle x2 +y2+6y+4=0 at the point pf intersection with the line 3x+y=2 are perpendicular to each other