Questions: Which of the following points lie on the unit circle? a. (4/5,-3/5) and (-7/25, 24/25) d. (4/5,-3/5) b. (-5/12,-12/13) e. DNE c. (-7/25, 24/25)

Transcript text: Which of the following points lie on the unit circle? a. $\left(\frac{4}{5},-\frac{3}{5}\right)$ and $\left(-\frac{7}{25}, \frac{24}{25}\right)$ d. $\left(\frac{4}{5},-\frac{3}{5}\right)$ b. $\left(-\frac{5}{12},-\frac{12}{13}\right)$ e. DNE c. $\left(-\frac{7}{25}, \frac{24}{25}\right)$

Solution

Solution Steps

Step 1: Understanding the Unit Circle

A point $(x, y)$ lies on the unit circle if it satisfies the equation $x^2 + y^2 = 1$.

Step 2: Checking Point $\left(\frac{4}{5}, -\frac{3}{5}\right)$

Substitute $x = \frac{4}{5}$ and $y = -\frac{3}{5}$ into the equation: \[ \left(\frac{4}{5}\right)^2 + \left(-\frac{3}{5}\right)^2 = \frac{16}{25} + \frac{9}{25} = \frac{25}{25} = 1. \] This point lies on the unit circle.

Step 3: Checking Point $\left(-\frac{7}{25}, \frac{24}{25}\right)$

Substitute $x = -\frac{7}{25}$ and $y = \frac{24}{25}$ into the equation: \[ \left(-\frac{7}{25}\right)^2 + \left(\frac{24}{25}\right)^2 = \frac{49}{625} + \frac{576}{625} = \frac{625}{625} = 1. \] This point also lies on the unit circle.

Final Answer

The points that lie on the unit circle are:
a. $\left(\frac{4}{5}, -\frac{3}{5}\right)$ and $\left(-\frac{7}{25}, \frac{24}{25}\right)$
d. $\left(\frac{4}{5}, -\frac{3}{5}\right)$
c. $\left(-\frac{7}{25}, \frac{24}{25}\right)$

Thus, the final answer is:
$\boxed{\text{a, c, d}}$

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