Normal

Q.

If the three normals drawn to the parabola, $y^2=2x$ pass through the point $\left(a,0\right),a\ne 0$, then ‘$a$’ must be greater than

1. $1$

2. $\frac{1}{2}$

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

4. $-1$

Q. The diameter of the circle whose centre lies on the line $x+y=2$ in the first quadrant and which touches both the lines $x=3$ and $y=2$ is

Q. The number of values of $c$ such that the straight line $y=2x+c$ touches the curve $4x^2+y^2=4$ is

1. $0$
2. $1$
3. $2$
4. infinite

Q. If the line $ax+by=0$ touches the circle $x^2+y^2+2x+4y=0$ and is a normal to the circle $x^2+y^2-4x+2y-3=0,$ then value of $(a,b)$ is/are

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

Q.

The equation of circle passing through $(4,5)$ and having the centre at $(2,2)$, is

1. $x^2+y^2+4x+4y-5=0$
2. $x^2+y^2-4x-4y-5=0$
3. $x^2+y^2-4x=13$
4. $x^2+y^2-4x-4y+5=0$

Q. The locus of point of intersection of two normals drawn to the parabola $y^2=4ax$ which are at right angles is

1. $y^2=a(x-3a)$
2. $y^2=a(x-a)$
3. $y^2=3a(x-2a)$
4. $y^2=2a(x-2a)$

Q. Which of the following line is a diameter of the circle $x^2+y^2-6x-8y-9=0$

1. 3x – 4y = 0
2. x + y = 7
3. x – y = 1
4. 4x – 3y = 9

Q. The equation of straight line passing through the point $(3,6)$ and cutting $y=\sqrt{x}$ orthogonally is

1. $4x+y=18$
2. $4x-y=6$
3. $x+y=9$
4. $x-y=9$

Q. If the normal to a parabola $y^2=4ax$ at $P$ meets the curve again at $Q$ and if $PQ$ and the normal at $Q$ makes angle $\alpha$ and $\beta$, respectively with the $x$-axis then $\tan \alpha (\tan \alpha +\tan \beta )$ has the value equal to

1. $0$
2. $-2$
3. $-\frac{1}{2}$
4. $-1$

Q. If the normal to the curve y = f(x) at x = 0 be given by the equation 3x – y + 3 = 0, then the value of $\underset{x\to 0}{lim}{\left\{f\right(x^2)-5f(4x^2)+4f(7x^2\left)\right\}}^{-1}$ is

1. $-\frac{1}{5}$
2. $-\frac{1}{4}$
3. $-\frac{1}{3}$
4. $-\frac{1}{2}$

Q. The angle between the normals to the parabola $y^2=24x$ at points $(6,12)$ and $(6,-12)$ is :

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. A curve $y=f\left(x\right)$ is passing through $(0,0).$ If the slope of the curve at any point $(x,y)$ is equal to $(x+xy),$ then the number of solution(s) of the equation $f\left(x\right)=1,$ is

1. $0$
2. $1$
3. $2$
4. $4$

Q. The equation of the normal to the circle $x^2+y^2=5$ at the point $(1,2)$ is

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

Q. The equation of the directrices of the hyperbola $3x^2-3y^2-18x+12y+2=0$ is

1. $x=3\pm \sqrt{\frac{13}{6}}$
2. $x=6\pm \sqrt{\frac{3}{13}}$
3. $x=3\pm \sqrt{\frac{6}{13}}$
4. $x=6\pm \sqrt{\frac{13}{3}}$

Q.

If the line x-y+k=0 is a normal to $y^2=4ax$ then the value of k is

1. 4a

2. -a

3. -5a

4. -3a

Q. Let us consider a curve, $y=f\left(x\right)$ passing through the point $(-2,2)$ and slope of the tangent to the curve at any point $(x,f(x\left)\right)$ is given by $f\left(x\right)+x{f}^{\prime }\left(x\right)=x^2$. Then

1. $x^3+xf\left(x\right)+12=0$
2. $x^2+2xf\left(x\right)+4=0$
3. $x^2+2xf\left(x\right)-12=0$
4. $x^3-3xf\left(x\right)-4=0$

Q. A line passes through (2, 0). The slope of the line, for which its intercept between y = x – 1 and y = –x + 1 subtends a right angle at the origin, is/are

1. 
2. 
3. 
4. 

Q. A straight line $L_1:x-2y+10=0$ meets the circle with equation $x^2+y^2=100$ at $B$ in the first quadrant. If another line $L_2$ through $B$ is perpendicular to $L_1$ cuts $y$-axis at $P(0,t)$, then the value of $t$ is

Q. If the equation $x^2+y^2-10x+21=0$ has real roots $x=\alpha$ and $y=\beta$ then

1. $3\le x\le 7$
2. $3\le y\le 7$
3. $-2\le y\le 2$
4. $-2\le x\le 2$

Q. The line $lx+my+n=0$ is normal to the circle $x^2+y^2+2gx+2fy+c=0,$ if

1. $lg+mf+n=0$
2. $lg+mf-n=0$
3. $lg-mf-n=0$
4. $lg-mf+n=0$

Q. From the point $P(1,5)$ tangents $PQ$ and $PR$ are drawn to the ellipse $\frac{x^2}{4}+\frac{y^2}{25}=1$, then the acute angle $QOR,$ where $O$ is the origin, is

1. ${\tan }^{-1}\frac{8}{15}$
2. ${\tan }^{-1}\frac{8}{35}$
3. ${\tan }^{-1}\frac{15}{8}$
4. ${\tan }^{-1}\frac{4}{15}$

Q. If $Ax+By=1$ is a normal to the curve $ay=x^2$ , then

1. $4A^2(1-aB)=a{B}^3$
2. $4A^2(1+aB)+a{B}^3=0$
3. $2A^2(2-aB)=a{B}^3$
4. $4A^2(2+aB)=a{B}^3$

Q. The line $ax+by+c=0$ is a normal to the circle $x^2+y^2=25.$ The portion of the line $ax+by+c=0$ intercepted by this circle is of length

Q. Let a curve $y=f\left(x\right)$ pass through the point $(2,({\log }_{e}2{)}^2)$ and have slope $\frac{2y}{x{\log }_{e}x}$ for all positive real value of $x.$ Then the value of $f\left(e\right)$ is equal to

Q.

Find the relation between $t_{1}and{t}_2$ if normals at $(a{t}_1^2,2a{t}_1)and(a{t}_2^2,2a{t}_2)$ meet on the parabola.

1. 

2. 

3. 

4. 

Q. Equation of the normal to $y^2=4x$ which is perpendicular to $x+3y+1=0$ is

1. $3x-y=6$
2. $3x-y=27$
3. $3x-y=21$
4. $3x-y=33$

Q. If equation of the plane through the straight line $\frac{x-1}{2}=\frac{y+2}{-3}=\frac{z}{5}$ and perpendicular to the plane $x-y+z+2=0$ is $ax-by+cz+4=0$, then the value of $a^2+b^2+c$ is

Q. An equation of the line that passes through $(10,-1)$ and is normal to $y=\frac{x^2}{4}-2$ is

Q. If the normals of the parabola $y^2=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x–3{)}^2+(y+2{)}^2=r^2,$ then the value of $r^2$ is

Q.

How do you find Maximum In Calculus?