Two parabolas y^2 = 4a(x - l_1) and x^2 = 4a( | vedclass

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(C) Let the point of contact be $(x_1, y_1)$.For the first parabola $y^2 = 4a(x - l_1)$,differentiating with respect to $x$ gives $2y \frac{dy}{dx} =

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$xy = 4a^2$

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(C) Let the point of contact be $(x_1, y_1)$.For the first parabola $y^2 = 4a(x - l_1)$,differentiating with respect to $x$ gives $2y \frac{dy}{dx} = 4a$,so $\frac{dy}{dx} = \frac{2a}{y_1}$.For the second parabola $x^2 = 4a(y - l_2)$,differentiating with respect to $x$ gives $2x = 4a \frac{dy}{dx}$,so $\frac{dy}{dx} = \frac{x_1}{2a}$.Since the parabolas touch at $(x_1, y_1)$,their slopes must be equal at this point.Therefore,$\frac{2a}{y_1} = \frac{x_1}{2a}$.This implies $x_1 y_1 = 4a^2$.Thus,the locus of the point of contact is $xy = 4a^2$.

List-$I$	List-$II$
$(I)$ Vertex	$(A)$ $(-\frac{3}{2}, -3)$
$(II)$ Focus	$(B)$ $(\frac{3}{2}, -3)$
$(III)$ Equation of the directrix	$(C)$ $2x+5=0$
$(IV)$ Equation of the axis	$(D)$ $2x+y+3=0$
	$(E)$ $y+3=0$
	$(F)$ $(-2, -3)$

Two parabolas $y^2 = 4a(x - l_1)$ and $x^2 = 4a(y - l_2)$ always touch one another,where $l_1$ and $l_2$ are variable. The locus of their point of contact has the equation:

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