Question

Slope Find all points on the circle \(x^{2}+y^{2}=25\) where the slope is \(\frac{3}{4}\).

Short answer

The points on the circle \(x^{2}+y^{2}=25\) where the slope of the tangent is \(\frac{3}{4}\) are approximately \((3,4)\), \((-3,-4)\), \((3,-4)\), and \((-3,4)\).

Step-by-step solution

Express Circle as a Function of x

Let's express the given circle \(x^{2}+y^{2}=25\) as a function of x. For the upper semicircle, we'll use \( y = \sqrt{25 - x^{2}}\) and for the lower semicircle, we'll use \( y = -\sqrt{25 - x^{2}}\).

Take Derivative to Find Slope

The slope of the tangent line to the circle at a given point is given by the derivative of the function. The derivative of the functions for the upper and lower semicircles can be calculated as follows: \( y'_{1} = \frac{-x}{\sqrt{25 - x^{2}}}\) for the upper semicircle and \( y'_{2} = \frac{x}{\sqrt{25 - x^{2}}}\) for the lower semicircle.

Find Points with Desired Slope

The desired slope is \(\frac{3}{4}\). Thus, by setting the derivatives obtained in the previous step equal to \(\frac{3}{4}\), we can find the x-coordinates where the slope of the tangent line has this value. Solving the equations \(\frac{-x}{\sqrt{25 - x^{2}}} = \frac{3}{4}\) and \(\frac{x}{\sqrt{25 - x^{2}}} = \frac{3}{4}\) will give us the values of x for the points we are looking for. Then we substitute x into the initial equation of the circle to find the corresponding y-values.

Solution

Solving the equations from the previous step, we find that the points on the circle where the slope is \(\frac{3}{4}\) are approximately \((3,4)\) and \((-3,-4)\) in the upper semicircle, and \((3,-4)\) and \((-3,4)\) in the lower semicircle.