The slope of the tangent to a curve at any point (x, y) on the curve is -x/y. Find the equation of the curve if the point (3, −4

Question

Helen [10]

4 years ago

10

The slope of the tangent to a curve at any point (x, y) on the curve is -x/y. Find the equation of the curve if the point (3, −4

) is on the curve.

Mathematics

2 answers:

natka813 [3]4 years ago

7 0

$\bf \stackrel{\textit{slope or derivative}}{-\cfrac{x}{y}}\qquad (\stackrel{x}{3},\stackrel{y}{-4})\qquad \left. -\cfrac{x}{y} \right|_{3,-4}\implies -\cfrac{3}{-4}\implies \stackrel{\stackrel{slope}{m}}{\cfrac{3}{4}}\\\\
-------------------------------\\\\
\stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-(-4)=\cfrac{3}{4}(x-3)
\implies 
y+4=\cfrac{3}{4}x-\cfrac{9}{4}
\\\\\\
y=\cfrac{3}{4}x-\cfrac{9}{4}-4\implies y=\cfrac{3}{4}x-\cfrac{25}{4}$

valkas [14] · Answer 1 · 2022-01-27T08:29:35+03:00

valkas [14]4 years ago

6 0

The differential equation
dy/dx = -x/y
is a separable differential equation that has the solution
x^2 + y^2 = c

Substituting the given values for x and y, we find c=25.

The desired equation is
x^2 + y^2 = 25