Standard Equation of Parabola

Q. The number of point(s) (x, y) (where x and y both are perfect squares of integers) on the parabola $y^2=px$, p being a prime number, is

1. zero
2. one
3. two
4. infinite

Q.

The parabola $y^2=x$ is symmetric about

1. x – axis

2. y – axis

3. Both x – axis and y – axis

4. The line y = x

Q. Consider the two curves
$C_1:y^2=4x,C_2:x^2+y^2-6x+1=0.$
Then,

1. $C_1$ and $C_2$ touch each other only at one point
2. $C_1$ and $C_2$ touch each other exactly at two points
3. $C_1$ and $C_2$ intersect (but do not touch) at exactly two points
4. $C_1$ and $C_2$ neither intersect nor touch each other

Q. The point P on the parabola $y^2=4ax$ for which $|PR-PQ|$ is maximum, where R = (-a, 0), Q = (0, a). is

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

Q. The vertex of the conic represented by $25(x^2+y^2)=(3x-4y+12{)}^2$ is

1. $(\frac{24}{25},-\frac{18}{25})$
2. $(0,0)$
3. $(-\frac{36}{25},\frac{48}{25})$
4. $(-\frac{18}{25},\frac{24}{25})$

Q. If point $(4,4)$ lies on a parabola whose focus lies on $x$ -axis and directrix is $lx+my=1$ such that $(l,m)$ lies on curve $x^2+y^2-32xy+8x+8y-1=0,$ then

1. Co-ordinates of focus may be $(5,0)$
2. Co-ordinates of focus may be $(3,0)$
3. If tangent at $(4,4)$ passes through origin, value of $m$ may be $\frac{1}{3}$
4. If tangent at $(4,4)$ passes through origin, equation of axis may be $x-4y-3=0$

Q. If one vertex of a chord of parabola $y^2=8ax$ is at $(0,0),$ then the locus of a point which divides the chord in the ratio $1:2$ is

1. $y^2=4ax$
2. $y^2=8ax$
3. $y^2=\frac{8a}{3}x$
4. $y^2=2ax$