Slope Form of Normal : Hyperbola

Q.

The solution set of the inequation 2x + y > 5 is

1. open half plane not containing the origin

2. half plane that contains the origin

3. whole xy-plane except the points lying on the line 2x + y = 5.

4. points on line

Q.

The equation of the normal to the curve x² = 4y which passes through the point (1, 2) is.

1. x + y - 3 = 0

2. 4x - y = 2

3. 4x - 3y + 2= 0

4. 4x - 2y = 0

Q. Assuming that the straight lines work as a plane mirror for a point find the image of the point (1, 2) in the line x- 3y + 4 = 0

Q.

The quadratic curve y = f (x) if it touches the line y = x at the point x = 1 and passes through the point (- 1, 0) is... .

1. 

2. 

3. 

4. 

Q.

The equation(s) of normals(s) to the curve 3x² - y² = 8 which is parallel to the line x + 3y = 4 is.

1. 3x - y + 8 = 0

2. x + 3y + 8 = 0

3. x + 3y + 3 = 0

4. x + 3y + 18 = 0

Q. Let $L$ be a normal to the parabola $y^2=4x$. If $L$ passes through the point $(9,6)$, then $L$ is given by

1. $y–x+3=0$
2. $y+3x-33=0$
3. $y–2x+12=0$
4. $y+x-15=0$

Q. Find the equation of normal to the curve $x^2=4y$ passing through the point $(1,2)$

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

Q. The equation of the normal to the curve $y^2=x^3$ at the point whose abscissa is $8$ is -

1. $x\pm \sqrt{2}y=104$
2. None of these
3. $x\pm 3\sqrt{2}y=104$
4. $3\sqrt{2}x\pm y=104$

Q. If $f\left(x\right)={\int }_1^x\frac{\ln t}{1+t}dt$ where $x>0$, then the value(s) of $x$ satisfying the equation, $f\left(x\right)+f\left(1/x\right)=2$ is

1. $2$
2. $e$
3. $e^2$
4. $e^{-2}$

Q. The family of curves passing through $(0,0)$ and satisfying the differential equation $\frac{y_2}{y_1}=k$, where $y_n=\frac{d^{n}y}{d{x}^n}$ is

1. $y=k$
2. $y=kx$
3. $y=k(e^x+1)$
4. $y=k(e^x-1)$

Q. The equation of the curve through the point $(1,0)$ and whose slope is $\frac{y-1}{x^2+x}$ is

1. $\left|\frac{(y-1)(x+1)}{x}\right|=2$
2. $2x(y-1)+x+1=0$
3. $x(y-1)(x+1)+2=0$
4. None of these

Q.

The equation of the normal of slope m to the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
is given by $y=mx\pm \frac{m(a^2+b^2)}{\sqrt{a^2-b^2{m}^2}}$

1. True

2. False

Q. $\int \frac{x^{6}dx}{1+x^7}$

1. $\frac{-1}{7}\ln (1+x^{-7})+C$
2. $\frac{1}{7}\ln (1+x^{-7})+C$
3. $\frac{-1}{7}\ln (1+x^7)+C$
4. $\frac{1}{7}\ln (1+x^7)+C$

Q. If ${\int }_{a-n}^n{e}^{x(a-x)}dx=\lambda$, then the value of ${\int }_a^{n}x{e}^{x(a-x)}dx,a\ne 2n$, is

1. $\frac{a\lambda }{2}$
2. $a\lambda$
3. None of these
4. $2a\lambda$

Q. $\mathrm{The}\mathrm{equation}\left(s\right)\mathrm{of}\mathrm{the}\mathrm{normal}\left(s\right)\mathrm{to}\mathrm{the}\mathrm{hyperbola}$ $3x^2-y^2=1\mathrm{having}\mathrm{slope}\frac{1}{3}$ $\mathrm{is}/\mathrm{are}:$

1. $\sqrt{2}(x-3y)=4$
2. $\sqrt{2}(x-3y)=-4$
3. $\sqrt{2}(x-3y)=8$
4. $\sqrt{2}(x-3y)=-8$

Q. Find the equation of a plane passing through the point (−1, −1, 2) and perpendicular to the planes 3x + 2y − 3z = 1 and 5x − 4y + z = 5.

Q. Number of real solution of the equation $\sqrt{{\log }_{10}(-x)}={\log }_{10}\sqrt{x^2}$ is

1. zero
2. exactly $2$
3. exactly $1$
4. $4$

Q. Assertion :The equation $3x^2-2y^2+4x-6y=0$ represents a hyperbola. Reason: The second degree equation $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a hyperbola if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0$ & $h^2>ab$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q. Find the equation of the curve which passes through the point (3, −4) and has the slope $\frac{2y}{x}$ at any point (x, y) on it.

Q. If the curve satisfying $ydx+(x+x^{2}y)dy=0$ passes through (1, 1), then $4\log y\left(4\right)-\frac{1}{y\left(4\right)}$ is equal to

1. 4
2. 8
3. 16
4. 3

Q. The normal to a given curve at each point (x, y) on the curve passes through the point (3, 0). If the curve contains the point (3, 4), find its equation.

Q. The equation of the curve passing through $(2,\frac{7}{2})$ and having gradient $1-\frac{1}{x^2}$ at a point $(x,y)$ is:

1. $y=x^2+x+1$
2. $xy=x^2+x+1$
3. $xy=x+5$
4. $xy=x^2+3$

Q. Q.45. Find a unit vector parallel to the xy plane and perpendicular to $4\hat{i}-3\hat{j}+\hat{k}$.

Q. The equation of normal to the curve $x^{\frac{2}{3}}$ + $y^{\frac{2}{3}}$ = $a^{\frac{2}{3}}$ at the point $(a,0)$ is -

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

Q. $\mathrm{The}\mathrm{equation}\left(s\right)\mathrm{of}\mathrm{the}\mathrm{normal}\left(s\right)\mathrm{to}\mathrm{the}\mathrm{hyperbola}$ $3x^2-y^2=1\mathrm{having}\mathrm{slope}\frac{1}{3}$ $\mathrm{is}/\mathrm{are}:$

1. $\sqrt{2}(x-3y)=4$
2. $\sqrt{2}(x-3y)=-4$
3. $\sqrt{2}(x-3y)=8$
4. $\sqrt{2}(x-3y)=-8$

Q. $\mathrm{The}\mathrm{equation}\left(s\right)\mathrm{of}\mathrm{the}\mathrm{normal}\left(s\right)\mathrm{to}\mathrm{the}\mathrm{hyperbola}$ $3x^2-y^2=1\mathrm{having}\mathrm{slope}\frac{1}{3}$ $\mathrm{is}/\mathrm{are}:$

1. $\sqrt{2}(x-3y)=4$
2. $\sqrt{2}(x-3y)=-4$
3. $\sqrt{2}(x-3y)=8$
4. $\sqrt{2}(x-3y)=-8$

Q.

The equation of the normal of slope m to the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
is given by $y=mx\pm \frac{m(a^2+b^2)}{\sqrt{a^2-b^2{m}^2}}$

1. True

2. False