Question

# Find the equations of the normal of the parabola  $${y^2} = 4x$$ which pass through the point $$\left( {3,0} \right)$$ .

Solution

## Here, $$y^2=4x$$, if compared with $$y^2=4ax$$ gives $$a=1$$For $$y^2=4ax$$, equation of normal is given by$$y=mx-2am-am^3$$Hence, for given parabola, equation of normal is$$y=mx-2m-m^3$$It passes through $$(3,0)$$$$\therefore 0=3m-2m-m^3$$$$\Rightarrow m^3-m=0 \Rightarrow m(m^2-1)=0$$$$\therefore m=0,1,-1$$.Hence equations of normals are:$$\Rightarrow y=0$$,$$\Rightarrow y=x-3$$,$$\Rightarrow y= -x+3$$ .Mathematics

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