Directrix

Q.

If the equation of the plane passing through the mirror image of a point $(2,3,1)$with respect to line $\frac{(x+1)}{2}=\frac{(y–3)}{1}=\frac{(z+2)}{-1}$ and containing the line $\frac{(x–2)}{3}=\frac{(1–y)}{2}=\frac{(z+1)}{1}$ is $\alpha x+\beta y+z=24$, then, $\alpha +\beta +\gamma$ is equal to:

1. $21$

2. $19$

3. $18$

4. $20$

Q. The vertex of the parabola $x^2+8x+12y+4=0$ is

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

Q.

Let the mirror image of the point $\left(1,3,a\right)$ with respect to the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$ be $\left(-3,5,2\right)$. Then, the value of $\left|a+b\right|$ is equal to ______.

Q.

The point on the curve $y=12x-x^3$ at which the gradient is zero are

1. $(0,2),(2,16)$

2. $(0,-2),(2,-16)$

3. $(2,-16),(-2,16)$

4. $(2,16),(-2,-16)$

Q. A line $L$ passing through the focus of the parabola $y^2=4(x-1),$ intersects the parabola at two distinct points. If $m$ be the slope of the line $L,$ then

1. $m\in R-\left\{0\right\}$
2. $-1<m<1$
3. $m<-1$ or $m>1$
4. None of these

Q. A parabola has the origin as its focus and the line $x=4$ as the directrix. Then the vertex of the parabola is at

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

Q. The equation of the directrix of the parabola whose vertex (3, 2) and focus (2, –1) is

1. x + 3y – 19 = 0
2. y - 2y – 9 = 0
3. 2x + 6y – 24 = 0
4. x – 3y – 19 = 0

Q. If x+k=0 is equation of directrix to parabola $y^2=8(x+1)$then k =

1. 1
2. 2
3. 3
4. 4

Q.

The locus of a point whose chord of contact with respect to parabola
$y^2=8x$ passes through focus is

1. $x+2=0$

2. Directrix of the given parabola

3. $x+1=0$

4. Latus ractum of the given parabola

Q. The mirror image of the directrix of the parabola $y^2=4(x+1)$ in the line mirror $x+2y=3$ is

1. $x=-2$
2. $4y+3x=16$
3. $3x-4y+16=0$
4. none of these

Q. If the focus of the parabola $(y-\beta {)}^2=4(x-\alpha )$ always lies between the lines $x+y=1$ and $x+y=3$ then

1. $1<\alpha +\beta <3$
2. $0<\alpha +\beta <3$
3. $0<\alpha +\beta <2$
4. $-1<\alpha +\beta <2$

Q. A line $L$ passing through the focus of the parabola $y^2=4(x-1),$ intersects the parabola at two distinct points. If $m$ be the slope of the line $L,$ then

1. $m\in R-\left\{0\right\}$
2. $-1<m<1$
3. $m<-1$ or $m>1$
4. None of these

Q. The equation of parabola, whose axis is parallel to $y-$axis and which passes through points $(0,2),(-1,0)$ and $(1,6)$ is

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

Q. If the focus is (1, –1) and the directrix is the line x + 2y – 9 = 0, the vertex of the parabola is

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

Q. If the line $x-1=0$ is the directrix of the parabola $y^2-kx+8=0,$ then values of $k$ are

1. $\frac{1}{8}$
2. $-8$
3. $4$
4. $\frac{1}{4}$

Q. If the line $x-1=0$ is the directrix of the parabola $y^2-kx+8=0,$ then values of $k$ are

1. $\frac{1}{8}$
2. $-8$
3. $4$
4. $\frac{1}{4}$

Q. If the focus is (1, –1) and the directrix is the line x + 2y – 9 = 0, the vertex of the parabola is

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

Q.

The Equation of the directrix to parabola $y^2=8x$ is _____

1. $x+2=0$

2. $x+3=0$

3. $x+1=0$

4. $x=0$

Q. A parabola has the origin as its focus and the line $x=4$ as the directrix. Then the vertex of the parabola is at

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

Q.

If the image of the point $\left(2,1\right)$ with respect to the line mirror be $\left(5,2\right)$, then the equation of the mirror is

1. $3x+y-12=0$

2. $3x-y+12=0$

3. $3x+y+12=0$

4. none of these