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