How to Find the Slope in a Circle

Use implicit differentiation to find a circle's slope.
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It is difficult to find the slope of a point on a circle because there is no explicit function for a complete circle. The implicit equation x^2 + y^2 = r^2 results in a circle with a center at the origin and radius of r, but it is difficult to calculate the slope at a point (x,y) from that equation. Use implicit differentiation to find the derivative of the circle equation to find the circle's slope.

    Find the equation for the circle using the formula (x-h)^2 + (y- k)^2 = r^2, where (h, k) is the point corresponding to the center of the circle on the (x, y) plane and r is the length of the radius. For example, the equation for a circle with its center at the point (1,0) and radius 3 units would be x^2 + (y-1)^2 = 9.

    Find the derivative of the above equation using implicit differentiation with respect to x. The derivative of (x-h)^2 + (y-k)^2 = r^2 is 2(x-h) + 2(y-k)dy/dx = 0. The derivative of the circle from step one would be 2x+ 2(y-1)*dy/dx = 0.

    Isolate the dy/dx term in the derivative. In the above example, you would have to subtract 2x from both sides of the equation to get 2(y-1)*dy/dx = -2x, then divide both sides by 2(y-1) to get dy/dx = -2x / (2(y-1)). This is the equation for the slope of the circle at any point on the circle (x,y).

    Plug in the x and y value of the point on the circle whose slope you wish to find. For example, if you wanted to find the slope at the point (0,4) you would plug 0 in for x and 4 in for y in the equation dy/dx = -2x / (2(y-1)), resulting in (-2_0) / (2_4) = 0, so the slope at that point is zero.

    Tips

    • When y = k, the equation has no solution (divide by zero error) because the circle has an infinite slope at that point.

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