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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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