The correct option is B x+3y−8=0
Let the reqiured normal(s) be drawn at a point P(a, b). The equation of the given curve is 3x2−y2=8 ......[1]
Differentiating both sides w.r.t. x, we get
6x−2ydydx=0⇒dydx=3xy⇒(dydx)(a,b)=3abSince, the normal at (a, b) is parallel to the linex+3y=4. Therefore,Slope of the normal at (a, b)=Slope of the line x+3y=4⇒−1(dydx)(a,b) =−13⇒−b3a=−13⇒a=b..(2]Since, (a, b) lies on (1]. Therefore,3a2−b2=8⇒3a2−a2=8 (From (2])⇒2a2=8⇒a2=4⇒a=±2If a=2, then b=2 and if a=−2, then b=−2Thus, the co−ordinates of the point are (2, 2)and (−2,−2). The equation of the normal at (2, 2) isy−2=−1(dydx)(a,b)(x−2)⇒y−2=−13(x−2)⇒x+3y−8=0The equation of the normal at (−2, −2) isy+2=−1(dydx)(a,b)(x+2)⇒y+2=−13(x+2)⇒x+3y+8=0.