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(A-(IV), B-(VI), C-(I), D-(II)) . The equation of the tangent to the parabola $y^2 = 4ax$ at $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$.For $y^2 = 4x$,$a =

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(A-(IV), B-(VI), C-(I), D-(II)) . The equation of the tangent to the parabola $y^2 = 4ax$ at $(x_1, y_1)$ is $yy_1 = 2a(x + x_1)$.For $y^2 = 4x$,$a = 1$. At $(2, \sqrt{8})$,the tangent is $y(\sqrt{8}) = 2(1)(x + 2) \Rightarrow \sqrt{8}y = 2x + 4 \Rightarrow 2\sqrt{2}y = 2x + 4 \Rightarrow x - \sqrt{2}y + 2 = 0$. Thus,$A ightarrow (iv)$.$B$. The equation of the normal to $y^2 = 4ax$ with slope $m$ is $y = mx - 2am - am^3$.For $y^2 = 16x$,$4a = 16 \Rightarrow a = 4$. The normal makes an angle of $45^{\circ}$ with the axis,so $m = 	an(135^{\circ}) = -1$ or $m = 	an(45^{\circ}) = 1$.For $m = 1$,$y = 1(x) - 2(4)(1) - 4(1)^3 = x - 8 - 4 = x - 12 \Rightarrow x - y - 12 = 0$. Thus,$B ightarrow (vi)$.$C$. For $y^2 = 12x$,$4a = 12 \Rightarrow a = 3$. The points on the parabola are $(at_1^2, 2at_1)$ and $(at_2^2, 2at_2)$.$A$ chord is a focal chord if $t_1 t_2 = -1$.Then $y_1 y_2 = (2at_1)(2at_2) = 4a^2(t_1 t_2) = 4(3)^2(-1) = 36(-1) = -36$. Thus,$C ightarrow (i)$.$D$. The equation $y^2 - kx + 16 = 0$ can be written as $y^2 = k(x - 16/k)$.Comparing with $Y^2 = 4AX$,we have $4A = k \Rightarrow A = k/4$.The directrix is $X = -A \Rightarrow x - 16/k = -k/4 \Rightarrow x = 16/k - k/4$.Given directrix is $x = 3$,so $16/k - k/4 = 3 \Rightarrow 64 - k^2 = 12k \Rightarrow k^2 + 12k - 64 = 0$.$(k + 16)(k - 4) = 0 \Rightarrow k = 4$ or $k = -16$. Thus,$D ightarrow (ii)$.

List-$I$	List-$II$
$A$. Equation of the tangent drawn at $(2, \sqrt{8})$ on the curve $y^2 = 4x$ is	$(i) -36$
$B$. Equation of the normal to the curve $y^2 = 16x$,that makes an angle of $45^{\circ}$ with its axis is	$(ii) 4$
$C$. The chord joining the points $(x_1, y_1)$ and $(x_2, y_2)$ on the curve $y^2 = 12x$ is a focal chord if $y_1 y_2 =$	$(iii) 8$
$D$. $A$ value of $k$ for which $x - 3 = 0$ is the directrix of the curve $y^2 - kx + 16 = 0$ is	$(iv) x - \sqrt{2}y + 2 = 0$
	$(v) x + y - 12 = 0$
	$(vi) x - y - 12 = 0$

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