Finding Roots of Quadratic

Q. The minimum distance between the curves $x^2+y^2+4x+16y+66=0$ and $y^2=8x$ is

1. $3\sqrt{2}$ units
2. $5\sqrt{2}$ units
3. $4\sqrt{2}-2$ units
4. $4\sqrt{2}+2$ units

Q. A line $L:y=mx+3$ meets $y$-axis at $E(0,3)$ and the arc of the parabola $y^2=16x,$ $0\le y\le 6$ at the point $F(x_0,y_0)$. The tangent to the parabola at $F(x_0,y_0)$ intersects the $y$-axis at $G(0,y_1)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $EFG$ has a local maximum
Match $\text{List I}$ with $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(P)}& m=& \left(1\right)& \frac{1}{2}\\ \text{(Q)}& \text{Maximum area of triangle EFG is}& \left(2\right)& 4\\ \text{(R)}& y_0=& \left(3\right)& 2\\ \text{(S)}& y_1=& \left(4\right)& 1\end{array}$

Which of the following is correct combination

1. $\left(P\right)\to \left(4\right),\left(Q\right)\to \left(1\right),\left(R\right)\to \left(2\right),\left(S\right)\to \left(3\right)$
2. $\left(P\right)\to \left(3\right),\left(Q\right)\to \left(4\right),\left(R\right)\to \left(1\right),\left(S\right)\to \left(2\right)$
3. $\left(P\right)\to \left(1\right),\left(Q\right)\to \left(3\right),\left(R\right)\to \left(2\right),\left(S\right)\to \left(4\right)$
4. $\left(P\right)\to \left(1\right),\left(Q\right)\to \left(3\right),\left(R\right)\to \left(4\right),\left(S\right)\to \left(2\right)$

Q. From the vertex of the parabola $y^2=4ax$ pair of perpendicular chords are drawn. If this chords are adjacent sides of a rectangle, then the locus of the vertex of the rectangle diagonally opposite to the vertex of parabola is

1. $y^2=4a(x-4a)$
2. $y^2=4a(x-6a)$
3. $y^2=4a(x-2a)$
4. $y^2=4a(x-8a)$

Q. A line $L:y=mx+3$ meets y –axis at $E(0,3)$ and the arc of the parabola $y^2=16x,0\le y\le 6$ at the point $F(x_0,y_0)$. The tangent to the parabola at $F(x_0,y_0)$ intersects the y-axis at $G(0,y_1)$. The slope m of the line L is chosen such that the area of the triangle EFG has a local maximum.
Match List-I with List-II and select the correct answer using the code given below the lists:

List - I	List - II
P. m =	1. $\frac{1}{2}$
Q. Maximum area of $\Delta EFG$ is	2. 4
R. $y_0=$	3. 2
S. $y_1=$	4. 1

1. P Q R S
   4 1 2 3
2. P Q R S
   1 3 2 4
3. P Q R S
   3 4 1 2
4. P Q R S
   1 3 4 2

Q. The area between the curves $y={\log }_{e}x$ and $y=({\log }_{e}x{)}^2$ is

1. $3-e$
2. $e-{\log }_{e}3$
3. $e-2$
4. $1$

Q. From the vertex of the parabola $y^2=4ax$ pair of perpendicular chords are drawn. If this chords are adjacent sides of a rectangle, then the locus of the vertex of the rectangle diagonally opposite to the vertex of parabola is

1. $y^2=4a(x-4a)$
2. $y^2=4a(x-6a)$
3. $y^2=4a(x-2a)$
4. $y^2=4a(x-8a)$