General Equation of Parabola

Q.

What is vertex of a quadratic equation? How do we find it?

Q. Consider a circle with its centre lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola. Then the point of intersection of the circle and parabola can be

1. $(-\frac{p}{2},p)$
2. $(\frac{p}{2},p)$
3. $(\frac{p}{2},-p)$
4. $(\frac{p}{2},-\frac{p}{2})$

Q. If the parabola $y=(a-b)x^2+(b-c)x+(c-a)$ touches the x - axis then the line ax + by + c = 0

1. Always passes through a fixed point
2. represents the family of parallel lines
3. always perpendicular to x-axis
4. always has negative slope

Q. $y=f\left(x\right)$ is the parabola of the form $y=x^2+ax+1,$ its tangent at the point of intersection of $y-$axis and parabola also touches the circle $x^2+y^2=r^2.$ It is known that no point of the parabola is below $x-$axis. The radius of the circle (in units) when $a$ attains its maximum value is

1. $\frac{1}{\sqrt{10}}$
2. $\frac{1}{\sqrt{5}}$
3. $1$
4. $\sqrt{5}$

Q.

How do you find the vertex of a quadratic equation?

Q. The point on the parabola $y^2=18x$, for which the ordinate is three times the abscissa, is

1. (6, 2)
2. (-2, -6)
3. (3, 18)
4. (2, 6)

Q. The locus of the midpoint of the focal distance of a variable point moving on the parabola $y^2=4ax$ is a parabola whose

1. Latus rectum is half the latus rectum of the original parabola
2. Vertex is $(\frac{a}{2},0)$
3. Directrix is $y-$axis
4. Focus has the co-ordinates $(a,0)$

Q. A parabolic curve is described parametrically by $x-3=t^2,$ $y=4t.$ Then equation of the parabola is

1. $y^2-4x+16=0$
2. $y^2-16x-48=0$
3. $y^2-16x+48=0$
4. $y^2=16x$

Q. A parabolic curve is described parametrically by $x-3=t^2,$ $y=4t.$ Then equation of the parabola is

1. $y^2-4x+16=0$
2. $y^2-16x-48=0$
3. $y^2-16x+48=0$
4. $y^2=16x$

Q. If equation of the parabola is $(6x+8y-5{)}^2=60(8x-6y+9),$ then

1. Equation of axis of the parabola is $6x+8y-5=0$
2. Equation of tangent at vertex of the parabola is $6x+8y-5=0$
3. Length of latus rectum is $\frac{3}{2}$ units
4. Length of latus rectum is $6$ units

Q.

Equation of the parabola whose axis is $y=x$, distance from origin to vertex is $\sqrt{2}$ and distance from origin to focus is $2\sqrt{2},$ is (Focus and vertex lie in 1st quadrant) :

1. $(x+y{)}^2=2(x+y-2)$

2. $(x-y{)}^2=8(x+y-2)$

3. $(x-y{)}^2=4(x+y-2)$

4. $(x+y{)}^2=4(x+y-2)$