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# P is a point, two tangents are drawn from it to the parabola $y^2=4x$ such that the slope of one tangent is three times the slope of the other. The locus of P is

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

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.
edited Apr 10, 2014 by meena.p