Position of a Point W.R.T Parabola

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. If $({\alpha }^2,\alpha -2)$ be a point interior to the regions of the parabola $y^2=2x$ bounded by the chord joining the points $(2,2)$ and $(8,-4),$ then $\alpha$ belongs to the interval

1. $-2+2\sqrt{2}<\alpha <2$
2. $\alpha >-2+2\sqrt{2}$
3. $\alpha >-2-2\sqrt{2}$
4. $\alpha <-2-2\sqrt{2}$

Q. The point $(-2m,m+1)$ is an interior point of the smaller region bounded by the circle $x^2+y^2=4$ and the parabola $y^2=4x$. Then $m$ belongs to the intyerval

1. $-5-2\sqrt{6}<m<1$
2. $0<m<4$
3. $-1<m<\frac{3}{5}$
4. $-1<m<-5+2\sqrt{6}$

Q.

The position of reflection of the point $\left(4,1\right)$ about the line$y=x-1$ is

1. $\left(1,2\right)$

2. $\left(3,4\right)$

3. $\left(-1,0\right)$

4. $\left(2,3\right)$

Q.

Which of the following points lie on the parabola $x^2=4ay?$

1. x=$a{t}^2$ , y=2at

2. x=2at , y=$a{t}^2$

3. x=2$a{t}^2$, y=at

4. x=2at, y=$a{t}^2$

Q. The mirror image of $y^2=4x$ about the line $x-y+1=0$ is

1. $(x-1{)}^2=4(y+1)$
2. $(x+1{)}^2=4(y+1)$
3. $(x+1{)}^2=4(y-1)$
4. $(x-1{)}^2=4(y-1)$

Q.

For $a>0$, let the curves $C_1:y^2=ax$ and $C_2:x^2=ay$ intersect at origin and a point .

Let the line $x=b(0<b<a)$ intersect the chord and the x-axis at points and , respectively.

If the line $x=b$ bisects the area bounded by the curves, $C_1$ and $C_2$ , and the area of $\Delta OQR=\frac{1}{2}$, then ‘$a$’ satisfies the equation

1. $x^6-12x^3+4=0$

2. $x^6-12x^3-4=0$

3. $x^6+6x^3-4=0$

4. $x^6-6x^3+4=0$

Q. The number of points with non-negative integral coordinates that lie in the interior of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=4x,$ is

Q. $\mathrm{The}\mathrm{value}\mathrm{of}a\mathrm{for}\mathrm{which}\mathrm{the}\mathrm{point}(a,2a)\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{interior}\mathrm{region}\mathrm{of}\mathrm{the}\mathrm{parabola}{y}^2=16x\mathrm{is}$
.

1. $0<a<4$
2. $0<a<8$
3. $-4<a<0$
4. $-8<a<0$

Q. The area of the region enclosed by the curves $y=x,x=e,y=\frac{1}{x}$ and the positive $x-$ axis is

1. $1$ square unit
2. $\frac{3}{2}$ square units
3. $\frac{5}{2}$ square units
4. $\frac{1}{2}$ square units

Q. The number of points with non-negative integral coordinates that lie in the interior of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=4x,$ is

1. $4$
2. $8$
3. $10$
4. $12$

Q. If $x+by+c=0$ is normal to the parabola $y^2=12x,$ tben complete set of all values of $c$ is

1. $(6,\infty )$
2. $(-\infty ,6)$
3. $(-\infty ,-6)\cup (6,\infty )$
4. $(-\infty ,\infty )-\left\{6\right\}$

Q. Set of values of $a$ for which point $P(2,4)$ lies inside the parabola $y^2=4ax$ is

1. $a\in (-\infty ,-2)$
2. $a\in (-2,\infty )$
3. $a\in (2,\infty )$
4. $a\in (-\infty ,2)$

Q. If $(\alpha +1,\alpha )$ be a point interior to the regions of the parabola $y^2=4x$ bounded by the chord joining the points $(6,5)$ and $(7,4)$, then the total number of integral values of $\alpha$ is

Q. If point $(k,k+2)$ lies inside the region bounded by parabolas $x^2=4(y+2)$ and $x^2=-(y-4)$ in first quadrant, then $k$ lies in the interval

1. $(0,2+2\sqrt{5})$
2. $(-2,0)$
3. $(0,1)$
4. $(2-2\sqrt{5},2+2\sqrt{5})$

Q.

The area of the figure bounded by the curves $y=\ln x$ and $y=(\ln x{)}^2$ is

1. $e+1$

2. $e-1$

3. $3-e$

4. $1$

Q. The point(s) which lies inside the region bounded by the curves $y^2=3x$ and $(x-2{)}^2=-4(y-4)$ is/are

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

Q. For $b>a>1,$ the area enclosed by the curve $y=\ln x,y-$axis and the straight lines $y=\ln a$ and $y=\ln b$ is

1. $b-a$
2. $b(\ln b-1)-a(\ln a-1)$
3. $(b-a)\ln a$
4. $\left(\ln b\right)\left(\ln a\right)$

Q.

If the co-ordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (-4, 3, -6) and (2, 9, 2) respectively, then the angle between the lines AB and CD is

1. 
   

2. 
   

3. 
   

4. None of these

Q. The area (in sq. units) of the region $A=\left\{\right(x,y):\frac{y^2}{2}\le x\le y+4\}$ is :

1. $18$
2. $30$
3. $16$
4. $\frac{53}{3}$

Q. Set of values of $\alpha$ for which the point $(\alpha ,1)$ lies inside the curves $c_1:x^2+y^2-4=0$ and $c_2:y^2=4x$ is

1. $|\alpha |<\sqrt{3}$
2. $\alpha >\frac{1}{4}$
3. $\frac{1}{2}<\alpha <\sqrt{2}$
4. $\frac{1}{4}<\alpha <\sqrt{3}$

Q. Consider the parabola, $(y+3{)}^2=2(x-5)$, then the correct option(s) among the following is/are

1. point $(5,-3)$ lies on the parabola
2. point $(2,3)$ lies inside the parabola
3. point $(0,4)$ lies outside the parabola
4. point $(8,-2)$ lies inside the parabola

Q. The mirror image of the directrix of the parabola
$y=x^2+5x+7$ in the line mirror $x+y=1$ is

1. $2y=1$
2. $2x-y=1$
3. $4x=y$
4. $x=\frac{1}{2}$

Q.

Which of the following can be the parametric equation of $x^2=4ay$

1. $(a{t}^2,2at)$

2. $(2at,a{t}^2)$

3. $(-2at,a{t}^2)$

4. $(a{t}^2,-2at)$

Q. The point $(-2m,m+1)$ is an interior point of the smaller region bounded by the circle $x^2+y^2=4$ and the parabola $y^2=4x$, then $m$ lies in the interval

1. $-1<m<-5+2\sqrt{6}$
2. $0<m<4$
3. $-5-2\sqrt{6}<m<1$
4. $-1<m<\frac{3}{5}$

Q. Let $\alpha$ & $\beta$ be the roots of the equation, $a{x}^2+bx+c=0$ where $1<\alpha <\beta$, then $\underset{x\to m}{lim}\frac{|a{x}^2+bx+c|}{a{x}^2+bx+c}=1$, then which of the following is correct?

1. $a>0$ & $m>1$
2. $a<0$ & $m<1$
3. $\frac{|a|}{a}=1$ & $m>\alpha$
4. $a<0$ & $\alpha <m<\beta$

Q. The equation of a parabola is $y^2=4x$. Let P (1, 3) and Q (1, 1) are two points in the xy plane. Then,

1. P and Q are exterior points
2. P is an interior point while Q is an exterior point
3. P and Q are interior points
4. P is an exterior point while Q is an interior point

Q. Area lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and the lines $x=0$ and $x=2$ is

1. $\frac{\pi }{2}$
2. $\pi$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. Area of the region bounded by $y=e^x,y=e^{-x}$ and the line $x=1$ is)

1. $e+\frac{1}{e}$
2. $e-\frac{1}{e}$
   
3. $e+e^{-1}+2$
   
4. $e+e^{-1}-2$

Q. Find image of a point $(-8,12)$ with respect to line mirror $4x+7y+13=0$.

1. $(-16,2)$
2. $(-16,-2)$
3. $(14,-2)$
4. $(-14,2)$