$\begin{array}{1 1}(A)\;\text{a straight line} \\(B)\;\text{a circle} \\(C)\;\text{a parapola} \\(D)\;\text{an ellipse} \end{array}$

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Let $P=(\alpha, \beta)$. Any tangent to the parabola is

$y= mx +\large\frac{a}{m}$

it passes through $(\alpha, \beta)$

So, $\beta =m \alpha +\large\frac{1}{m}$

$\therefore m^2 \alpha=- \beta m+1=0$

Its roots are $m_1,3m_1$

So, $m_1.3m_1=\large\frac{1}{\alpha}$

$\therefore 3. \bigg( \large\frac{\beta}{4 \alpha} \bigg)^2=\large\frac{1}{\alpha}$

or $3 \beta^2=16 \alpha$

Thus the locus is $3y^2=16x$ which is a parabola.

Hence C is the correct answer.

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