Focus

Q. If $(2,0)$ is vertex and $y-$axis is the directrix of a parabola. Then its focus will be

1. $(0,0)$
2. $(2,0)$
3. $(4,0)$
4. $(-2,0)$

Q. The extremities of latus rectum of a parabola are $(1,1)$ and $(1,-1)$. Then the equation(s) of the parabola is/are

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

Q.

The coordinates of the focus of the parabola $y^2-x-2y+2=0$ are

1. $(\frac{5}{4},1)$

2. $(\frac{1}{4},0)$

3. (1, 1)

4. none of these

Q. The focus of the parabola $y^2=4y-4x$ is

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

Q.

The focus of the parabola y=$2x^2+x$ is

1. $(\frac{-1}{4},0)$

2. $(\frac{-1}{4},\frac{1}{8})$

3. (0, 0)

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

Q. The length of latus rectum of parabola $4y^2+3x+3y+1=0$ is

1. $\frac{4}{3}$
2. $7$
3. $12$
4. $\frac{3}{4}$

Q. If the focus of the parabola $x^2-ky+3=0$ is $(0,2)$, then the value(s) of $k$ is/are

1. $4$
2. $6$
3. $3$
4. $2$

Q.

The latus rectum of parabola $y^2=5x+4y+1$ is

1. 

2. 

3. 

4. 

Q. The equation of reflection of $y^2=x$ about $y-$axis is

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

Q. The locus of the centre of a circle passing through a point and touching a line is

1. a parabola
2. a hyperbola
3. an ellipse
4. a straight line

Q. Co-ordinate of the focus of the parabola $x^2-4x-8y-4=0$ are

1. 
2. 
3. 
4. 

Q. The parabola $y^2=px$ passes through the point of intersection of lines $\frac{x}{3}+\frac{y}{2}=1$ and $\frac{x}{2}+\frac{y}{3}=1$ its focus is

1. $(\frac{3}{10},0)$
2. $(\frac{3}{5},0)$
3. $(\frac{3}{7},0)$
4. $(\frac{6}{7},0)$

Q. From any point P(h, k), four normal can be drawn to the rectangular hyperbola $xy=c^2$ such that product of the abscissae of the feet of the normal = product of the ordinates of the feet of the normal is equal to

1. 
2. 
3. 
4. 

Q. The end points of the latus rectum of a parabola having vertex at origin are $(4,8)$ and $(4,-8)$ then the equation of the parabola is

1. $y^2=4x$
2. $y^2=8x$
3. $y^2=16x$
4. $x^2=8y$

Q.

The focus of the parabola $x^2-2x=2y$ is

1. 

2. 

3. 

4. 

Q. Let $P\left(x\right)$ and $Q\left(x\right)$ be two polynomials. Suppose that $f\left(x\right)=P\left(x^3\right)+xQ\left(x^3\right)$ is divisible by $x^2+x+1$, then

1. $P\left(x\right)$ is divisible by $(x-1)$, but $Q\left(x\right)$ is not divisible by $(x-1)$
2. $Q\left(x\right)$ is divisible by $(x-1)$, but $P\left(x\right)$ is not divisible by $(x-1)$
3. Both $P\left(x\right)$ and $Q\left(x\right)$ are divisible by $(x-1)$
4. $f\left(x\right)$ is divisible by $(x^3-1)$

Q. The coordinates of focus of parabola $9y^2-9x-12y-57=0$ is

1. $(-\frac{235}{36},-\frac{2}{3})$
2. $(-\frac{235}{36},\frac{2}{3})$
3. $(\frac{235}{36},\frac{2}{3})$
4. $(\frac{235}{36},-\frac{2}{3})$

Q. If $(2,0)$ is vertex and $y-$axis is the directrix of a parabola. Then its focus will be

1. $(0,0)$
2. $(2,0)$
3. $(4,0)$
4. $(-2,0)$

Q. The focus of the parabola $y^2=4y-4x$ is

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

Q.

The latus rectum of parabola $y^2=5x+4y+1$ is

1. $\frac{5}{4}$

2. $10$

3. $5$

4. $\frac{5}{2}$

Q. value of x satisfying the equality $|x^2+8x+7|=|x^2+4x+4|+|4x+3|$ for $x\in R$ are

1. $(\frac{3}{4},\infty )\cup \{-2\}$
2. $[-\frac{3}{4},\infty )\cup \{-2\}$
3. $(-2,\infty )$
4. $(-\frac{4}{3},\infty )$

Q. The coordinates of the focus of the parabola described parametrically by $x=5t^2+2$ and $y=10t+4$ are

1. $(7,4)$
2. $(3,4)$
3. $(3,-4)$
4. $(-7,4)$

Q. Let $P$ be the point $(1,0)$ and $Q$ be a point on the curve $y^2=8x$. Then the locus of the mid-point of $PQ$ is

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

Q. A ray of light travels along a line $y=4$ and strikes the surface of a curves $y^2=4(x+y)$, then equation of the line along which reflected ray travels is

1. $2x+y=4$
2. $x+y=4$
3. $x=0$
4. $x=2$

Q. Equation $x^2-2x-2y+5=0$ represents

1. a parabola with vertex $(1,2)$
2. a parabola with directrix $y=\frac{2}{5}$
3. a parabola with vertex $(2,1)$
4. a parabola with directrix $y=\frac{3}{2}$

Q. Find the vertex, axis, latus rectum, and focus of the parabolas $x^2+2y=8x-7.$

Q. Consider a circle with center lying on the focus of the parabola $y^2=2px$ such that it touches the directrix of the parabola, then a point of intersection of the circle and the parabola is

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

Q. Find the equation of the circle passing through the points $(1,1),(-1,-2)$ and whose centre lies on the line $x+2y=1$

Q. The parabola $y^2=px$ passes through the point of intersection of lines $\frac{x}{3}+\frac{y}{2}=1$ and $\frac{x}{2}+\frac{y}{3}=1$ its focus is

1. $(\frac{3}{10},0)$
2. $(\frac{3}{5},0)$
3. $(\frac{3}{7},0)$
4. $(\frac{6}{7},0)$

Q. Let P be the point (1, 0) and Q a point on the curve $y^2=8x$. The locus of mid point of PQ is-

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