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(D) The equation of the parabola is $y^2 = 4x$. Comparing this with $y^2 = 4ax$,we get $a = 1$.The slope of the tangent $m = 	an(45^{\circ}) = 1$.The

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None of these

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(D) The equation of the parabola is $y^2 = 4x$. Comparing this with $y^2 = 4ax$,we get $a = 1$.The slope of the tangent $m = 	an(45^{\circ}) = 1$.The equation of a tangent to the parabola $y^2 = 4ax$ with slope $m$ is $y = mx + \frac{a}{m}$.Substituting $a = 1$ and $m = 1$,we get $y = 1(x) + \frac{1}{1}$,which simplifies to $y = x + 1$.The point of contact $(x_1, y_1)$ for a tangent with slope $m$ to the parabola $y^2 = 4ax$ is given by $(\frac{a}{m^2}, \frac{2a}{m})$.Substituting $a = 1$ and $m = 1$,we get the point of contact $(x_1, y_1) = (\frac{1}{1^2}, \frac{2(1)}{1}) = (1, 2)$.Since the point of contact is a fixed point $(1, 2)$,the locus is simply the point itself,which does not match any of the given algebraic expressions.

Find the locus of the point of contact of the tangent to the parabola $y^2 = 4x$,where the tangent makes an angle of $45^{\circ}$ with the $x$-axis.

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