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(D) The equation of the parabola is $x^2 = y$,which is of the form $x^2 = 4ay$ with $4a = 1$,so $a = 1/4$.The equation of a tangent with slope $m$ is

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$24$

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(D) The equation of the parabola is $x^2 = y$,which is of the form $x^2 = 4ay$ with $4a = 1$,so $a = 1/4$.The equation of a tangent with slope $m$ is $y = mx - am^2$,which becomes $y = mx - \frac{m^2}{4}$.If this tangent passes through a point $(h, k)$,then $k = mh - \frac{m^2}{4}$,which simplifies to $m^2 - 4hm + 4k = 0$.Let $m_1$ and $m_2$ be the slopes of the two tangents from $(h, k)$. Then $m_1 + m_2 = 4h$ and $m_1m_2 = 4k$.The angle between the tangents is $45^{\circ}$,so $	an 45^{\circ} = |\frac{m_1 - m_2}{1 + m_1m_2}| = 1$.Thus,$(m_1 - m_2)^2 = (1 + m_1m_2)^2$.Using $(m_1 - m_2)^2 = (m_1 + m_2)^2 - 4m_1m_2$,we get $(4h)^2 - 4(4k) = (1 + 4k)^2$.$16h^2 - 16k = 1 + 8k + 16k^2$.Replacing $(h, k)$ with $(x, y)$,we get $16x^2 - 16y = 1 + 8y + 16y^2$,or $16y^2 - 16x^2 + 24y + 1 = 0$.Comparing this with $16y^2 - 16x^2 + ky + 1 = 0$,we find $k = 24$.

Let the locus of the points from which the tangents drawn to $y = x^2$ make an angle of $45^{\circ}$ with each other be $16y^2 - 16x^2 + ky + 1 = 0$. Then $k$ is equal to:

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