Question

using the following equation, find the center and radius: ( x^2 + 2x + y^2 + 4y = 20 )
the center is located at (1, 2), and the radius is 25
the center is located at (-1, 2), and the radius is 25
the center is located at (-1, -2), and the radius is 5
the center is located at (1, 2), and the radius is 5
question 2 (multiple choice worth 1 point)
lines cd and de are tangent to circle a, as shown below:
image of circle a with tangents cd and de from point d, arc ce is 105°
if ( overset{\frown}{ce} ) is 105°, what is the measure of ( angle cde )?
52.5°
62.5°
75°
255°

Explanation:

First Question (Circle Equation)

Step1: Group x and y terms

$x^2 + 2x + y^2 + 4y = 20$

Step2: Complete the square for x

$x^2 + 2x + 1 = (x+1)^2$, add 1 to right: $20+1=21$

Step3: Complete the square for y

$y^2 + 4y + 4 = (y+2)^2$, add 4 to right: $21+4=25$

Step4: Write standard circle form

$(x+1)^2 + (y+2)^2 = 5^2$

Step1: Find central angle supplement

Major arc $\overset{\frown}{CBE} = 360^\circ - 105^\circ = 255^\circ$

Step2: Use tangent angle formula

$\angle CDE = \frac{1}{2}(\text{Major arc } \overset{\frown}{CBE} - \text{Minor arc } \overset{\frown}{CE})$

Step3: Calculate angle value

$\angle CDE = \frac{1}{2}(255^\circ - 105^\circ) = \frac{1}{2}(150^\circ) = 75^\circ$

Answer:

The center is located at (-1, -2), and the radius is 5

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Second Question (Circle Tangents & Arcs)