(Myanmar Examboard Matriculation, 2019 No 14 (b))
Find the normals to the curve $ x y+2 x-y=0 $ that are parallel to the line $2 x+y=0 .$ [5 Marks]Solution:
\begin{eqnarray*}x y+2 x-y&=&0 \\y(x-1) &=&-2 x \\y &=&\dfrac{-2 x}{x-1} =-2-\frac{2}{x-1}=-2-2(x-1)^{-1}\\\dfrac{d y}{d x} &=&0+2(x-1)^{-2} =\dfrac{2}{(x-1)^{2}}\end{eqnarray*}
Since gradient of normal is $-2,\dfrac{d y}{d x}=\dfrac{1}{2}.$
Thus,
\begin{eqnarray*}\frac{2}{(x-1)^{2}} &=&\frac{1}{2} \\(x-1)^{2} &=&4 \\x-1 &=&\pm 2 \\x &=&-1 \text { or } x=3 . \end{eqnarray*}
If $x=-1, y=-1.$
At $(-1,-1)$ the normal equation is $ y+1=-2(x+1) .$
If $x=3, y=-3.$
At $(3,-3)$ the normal equation is $y+3=-2(x-3).$
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