Orthogonal Trajectories

Q. Find the orthogonal trajectory of $x^2+y^2=c$

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

Q. $\mathrm{The}\mathrm{orthogonal}\mathrm{trajectory}\mathrm{of}{y}^2=4\mathrm{ax}\mathrm{is}$.

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

Q. A particle moves along the curve $6y=x^3+2$, Find the points on the curve at which the $y-$coordinate is changing $8$ times as fast as the $x-$coordinate.

Q.

Find the orthogonal trajectory of $x^2+y^2=c$

1. $y=x^2$

2. $x=y^2$

3. $y=mx$

4. $x^2-y^2=c$

Q. Can 3 vectors not lying in a plane never add up to give a null vector? Why?

Q. The orthogonal trajectories of the family of curves $a^{n-1}y=x^n$ are given by

1. $x^n+n^{2}y=$ constant
2. $n{y}^2+x^2=$ constant
3. $n^{2}x+y^n=$ constant
4. $n^{2}x-y^n=$ constant

Q. the differential equation of all conics whose axes coicide with the coordinate axes i

Q. Angle between orthogonal trajectory and normals of a given family of curves (in degrees) at a given point on the curve = ___

Q. Consider a curve passing through $(1,1)$ such that perpendicular distance of normal drawn at any point $P$ from origin is equal to ordinate of the point $P.$ Then which of the following statement(s) is/are correct?

1. The differential equation for the curve is $\frac{dy}{dx}=\frac{x^2-y^2}{2xy}.$
2. The differential equation for the curve is $\frac{dy}{dx}=\frac{y^2-x^2}{2xy}.$
3. The equation of tangent at $(2,0)$ is $x=2.$
4. The curve passes through origin.

Q. The family of orthogonal trajectories of $xy=C_1$ is:
(where $C_1,C_2$ are arbitrary constants)

1. $2x^2-y^2=C_2$
2. $x^2-2y^2=C_2$
3. $x^2-y^2=C_2$
4. $x^2+y^2=C_2$

Q. What is the orthogonal trajectory of $x^2-y^2=c$?

1. $x^2+y^2=c$
2. $\mathrm{xy}=c$
3. $y=c$
4. $3x^2+y^2=c$

Q.

Formation of the differential equation corresponding to the ellipse major axis 2a and minor axis 2b is:

1. 

2. 

3. 

4. 

Q. Find the differential equation of the family of { circles having their centres on the line y=10 and touching the X-axis.

Q. The differential equation for a family of curves is $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y}{2x}.$ What is the differential equation for the orthogonal trajectory of the curves?

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

Q. $\mathrm{The}\mathrm{orthogonal}\mathrm{trajectory}\mathrm{of}{y}^2=4\mathrm{ax}\mathrm{is}$.

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

Q. The orthogonal trajectories of the family of curves $a^{n-1}y=x^n$ are given by

1. $x^1+n^{2}y=$ constant
2. $n{y}^2+x^2=$ constant
3. $n^{2}x+y^n=$ constant
4. $n^{2}x-y^n=$ constant

Q. The orthogonal trajectories of the family of curves $a^{n-1}y=x^n$ are given by

1. $x^n+n^{2}y$ =constant
   
2. $n{y}^2+x^2$ =constant
   
3. $n^{2}x+y^n$=constant
   
4. $n^{2}x-y^n$ =constant

Q. $\mathrm{The}\mathrm{orthogonal}\mathrm{trajectories}\mathrm{of}\mathrm{the}\mathrm{line}y=2\mathrm{include}$

1. x=2
2. x=3
3. x=1
4. All of the above

Q. At what points of curve $y=\frac{2}{3}{x}^3+\frac{1}{2}{x}^2$, the tangent makes equal angle with the axis

1. $\left(\frac{1}{2},\frac{5}{24}\right)$ and $(-1,-\frac{1}{6})$
2. $\left(\frac{1}{3},\frac{1}{7}\right)$and $(-3,\frac{1}{2})$
3. $\left(\frac{1}{2},\frac{4}{9}\right)$ and $(-1,0)$
4. $\left(\frac{1}{3},\frac{4}{47}\right)$ and $(-1,-\frac{1}{3})$

Q. Write the differential equation representing family of curves y = mx, where m is arbitrary constant.

Q. What is the orthogonal trajectory of $x^2-y^2=c$?

1. $x^2+y^2=c$
2. $\mathrm{xy}=c$
3. $y=c$
4. $3x^2+y^2=c$

Q. The orthogonal trajectories of the family of curves $a^{n-1}y=x^n$ are given by

1. $x^n+n^{2}y$ =constant
   
2. $n{y}^2+x^2$ =constant
   
3. $n^{2}x+y^n$=constant
   
4. $n^{2}x-y^n$ =constant

Q.

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

Q. $\mathrm{The}\mathrm{orthogonal}\mathrm{trajectories}\mathrm{of}\mathrm{the}\mathrm{line}y=2\mathrm{include}$

1. x=2
2. x=3
3. x=1
4. All of the above

Q. ntA particle projected from origin in x-y plane with a velocity v vector =3i+6xj, where i and j are unit vectors along x and y axis .Find the equation of path followed by the particle .n

Q. The differential equation of all straight lines in a plane passing through (0, 1) is:

1. $y=m(x-1)$
2. $y-1=mx$
3. $y=x{y}_1$
4. $y=x{y}_1+1$

Q. Find the differential equation of all non horizontal line in a plane.

Q. The differential equation for a family of curves is $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y}{2x}.$ What is the differential equation for the orthogonal trajectory of the curves?

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

Q. Angle between orthogonal trajectory and normals of a given family of curves (in degrees) at a given point on the curve = ___

Q. The orthogonal trajectories of the family of curves $a^{n-1}y=x^n$ are given by

1. $x^n+n^{2}y=$ constant
2. $n{y}^2+x^2=$ constant
3. $n^{2}x+y^n=$ constant
4. $n^{2}x-y^n=$ constant