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.

How do you know if a hyperbola is open?

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.

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

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

Q. 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 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 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. $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{y}{x}$