equation of tangent

find the equation for the tangent to y = x^4 at the point where x=2
first substitute x=2 into the given equation to find y to get the point (x1,y1)
at which the tangent is required.
find dy/dx and evaluate it at (x1,y1) to get the slope,m of the tangent
then use (y-y1) = m(x-x1) to get the equation of the tangent

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orthogonal trajectory of y = (k/x)

find the orthogonal trajectory of y = (k/x)

first we find the diff. equation of the given family
y=(k/x) -----------(1)
diff. w.r.t. x
y' = -k / (x²)---------(2)

eliminating k , divide (2) by (1)

y' / y = -1/x ---------(3)

(3) is the diff. equation of the given family.
replace y with -1/y' in (3) to get the diff. equation of the orthogonal trajectory

diff. equation of the orthogonal trajectory is

-yy' = -1/x
y(dy/dx) = 1/x

ydy =xdx
integrating and simplifying, we see that the orthogonal trajectory is

x² - y² = C


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