Find the orthogonal trajectories for the family of curves y=(kx)^6.
for our function,
y' = 6k(kx)^5
so, we want other curves whose slope is
y' = -1/(6k^6) x^-5
y = 1/(24k^6 x^4)
or, to make things look more similar,
y = 1/(24k^2) (kx)^-4
To find the orthogonal trajectories for the family of curves y = (kx)^6, we follow these steps:
Step 1: Calculate the derivative of y = (kx)^6 with respect to x.
dy/dx = 6(kx)^5
Step 2: Find the slope of the tangent line to any curve in the family of curves y = (kx)^6. This slope is given by dy/dx = 6(kx)^5.
Step 3: Find the negative reciprocal of the slope obtained in step 2 to get the slope of the orthogonal trajectory. The negative reciprocal of 6(kx)^5 is -1/(6(kx)^5) or -1/6(kx)^5.
Step 4: Now, we have the slope of the orthogonal trajectory. We can write it as dy/dx = -1/6(kx)^5.
Step 5: Solve the differential equation dy/dx = -1/6(kx)^5 for the orthogonal trajectories. To do this, we separate the variables and integrate:
∫1dy = ∫-1/6(kx)^5dx
Step 6: Integrating both sides gives:
y = -1/30k^5∫x^5dx
Step 7: Integrating further gives:
y = -1/30k^5 * (1/6)x^6 + C
This equation represents the family of orthogonal trajectories for the given family of curves y = (kx)^6, where C is the constant of integration for each orthogonal trajectory.
Therefore, the orthogonal trajectories for the family of curves y = (kx)^6 are given by the equation y = -1/30k^5 * (1/6)x^6 + C.
To find the orthogonal trajectories for the family of curves y = (kx)^6, we can follow these steps:
Step 1: Find the derivative of the given family of curves with respect to x.
Take the derivative of y = (kx)^6 with respect to x using the power rule of differentiation:
dy/dx = 6(kx)^5 * k
dy/dx = 6k^6x^5
Step 2: Determine the slope of the tangent line for the given family of curves at each point.
The slope of the tangent line for the family of curves is given by 6k^6x^5.
Step 3: Find the negative reciprocal of the slope obtained in Step 2 to obtain the slope of the orthogonal trajectories.
Let m be the slope of the orthogonal trajectories. Then, we have:
m = -1 / (6k^6x^5)
Step 4: Solve the equation obtained in Step 3 to find the equation of the orthogonal trajectories.
Using the slope-point form of a line, given a point (x, y), we have:
(y - y1) = m(x - x1)
(y - (kx)^6) = (-1 / (6k^6x^5)) * (x - x1)
Simplify the equation and rearrange it to find the equation of the orthogonal trajectories.
Please note that this process produces a differential equation that can be used to find the equation of the orthogonal trajectories. Solving this differential equation will provide the specific equations for the orthogonal trajectories.