Question

The equation y - 2x = c represents the orthogonal trajectories of the family of curves given
by
Select one:
O a. 2r +y=C
O b. 2 - 2y = C
O c. 2r – y=C
O d. 2 + 2y = C
ا​

Answer 1

Answe

y=Cx 2

⇒ x 2

y=C,

Differentiating

x4x 2y −2xy

=0

⇒y

= x2y

Replacing

dx dd

by − dydx

, we have y

′

= 2y−x

⇒2ydy+xdx=0

⇒y 2 + 21 x 2

=constant

which represent a family of ellipses.

Answer 2

Correct Question:

The equation of the orthogonal trajectories of the family of curves that the equation $y - 2x = c$ represents is given by $y=2x+C$ and the correct answer is option 1.

Select one

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

Final Answer:

The equation of the orthogonal trajectories of the family of curves that the equation $y - 2x = c$ represents is

Given:

The equation is given as $y - 2x = c$, and the equation which represents the orthogonal trajectories of the family of curves is given in the options.

To Find:

It is required to find the equation of the orthogonal trajectories of the family of curves that the equation $y - 2x = c$ represents.

Explanation:

On performing the differentiation with respect to one independent variable, the differentiation of the dependent variable is obtained.

On performing integration, a constant term known as the integrational constant is obtained.

Step 1 of 3

Differentiating both sides of the equation $y - 2x = c$ , we get the following.

$\frac{d}{dx} (y - 2x) = \frac{d}{dx}c\\\frac{dy}{dx}-2=0\\\frac{dy}{dx}=2\\dy=2dx$

Step 2 of 3

Integrating both sides of the equation $\frac{dy}{dx}=2$ , we get the following.

$\int\limits dy=2\int\limits dx\\y=2x+C$

Step 3 of 3

Among the following options

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

The correct option corresponds to $y=2x+C$.

Therefore, the required equation of the orthogonal trajectories of the family of curves that the equation $y - 2x = c$ represents is given by $y=2x+C$ and the correct answer is option 1.

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