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  • The number of distinct normals that can be drawn from <img alt="\mathrm{\left(\frac{11}{4}, \frac{1}{4}\right) }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cleft%28%5C

The number of distinct normals that can be drawn from \mathrm{\left(\frac{11}{4}, \frac{1}{4}\right) }to the parabola \mathrm{ y^2=4 x} is 

 

Option: 1

3


Option: 2

2


Option: 3

1


Option: 4

4


Answers (1)

best_answer

For \mathrm{y^2=4 x, a=1.}

The equation of normal at \mathrm{\left(a t^2, 2 a t\right) \equiv\left(t^2, 2 t\right)} is

            \mathrm{ y-2 t=-t\left(x-t^2\right) }

It is drawn from \mathrm{A\left(\frac{11}{4}, \frac{1}{4}\right)}. The condition is

              \mathrm{ \frac{1}{4}-2 t=-t\left(\frac{11}{4}-t^2\right) }

Or \mathrm{\quad t^3-\frac{11}{4} t+2 t-\frac{1}{4}=0}

Or \mathrm{\quad t^3-\frac{3}{4} t-\frac{1}{4}=0}

\mathrm{ \begin{aligned} & \text { Or } \quad 4 t^3-3 t-1=0 \\\\ & (2 t+1)\left(2 t^2-t-1\right)=0 \\\\ & \Rightarrow \quad t=-\frac{1}{2}, \frac{1 \pm \sqrt{1+8}}{4} \\\\ & =-\frac{1}{2}, 1,-\frac{1}{2}=-\frac{1}{2},-\frac{1}{2}, 1 \\ & \end{aligned} }
\therefore \quad  The points of contact are

\mathrm{ \left(\frac{1}{4},-1\right),\left(\frac{1}{4},-1\right),(1,2) }

Two distinct normals as two points of contact are coincident.

Posted by

shivangi.shekhar

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