Intersection of a Line and Finding Roots of a Parabola

Q. If the line y = x + 2a touches the parabola $y^2=4a(x+a)$, then the point of contact is .

1. (0, a)
2. (0, 2a)
3. (a, 0)
4. (2a, 0)

Q.

The straight line $y=2x+\lambda$ does not meet the parabola $y^2=2x$, if

1. $\lambda <\frac{1}{4}$

2. $\lambda >\frac{1}{4}$

3. $\lambda =4$

4. $\lambda =1$

Q. Locus of the point z satisfying the equation |iz - 1| + |z - i| = 2 is

Q.

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is___

Q. A line $x-\frac{y}{\sqrt{3}}=c$ is drawn through the focus $\left(F\right)$ of the parabola $y^2-8x-16=0.$ If the two intersection points of the given line and the parabola are $P$ and $Q$, such that the perpendicular bisector of $PQ$ intersects the $x$-axis at $A$, then the length of $AF$ is

1. $8$
2. $\frac{8}{3}$
3. $\frac{16}{3}$
4. $\frac{8}{7}$

Q. The equation of the tangent to the parabola y = $x^2$ - 2x + 3 at the point (2, 3) is .

1. x = 4y - 1
2. x = 2y - 1
3. y = 4x - 1
4. y = 2x - 1

Q. the line L1 passing through the point(1, 1) and the line L2 passes through the point(-1, 1) .If the differenceof the slope of the lines is 2.Find the locus of the point of intersection of the line L1 and L2

Q. Let S be the focus of the parabola $y^2=8x$ and $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the $\Delta OPS$ ($O$ is the origin) is ___.