Let's go through the questions from the image you uploaded one by one, solving them step by step.

Question 1

The circle $C$ has the equation: $x^2 + y^2 - 2x + 6y = 26$ We need to:

1. Find the coordinates of the center of C.
2. Find the radius of C.

Step 1: Rearranging the equation into standard form

We will complete the square for both $x$ and $y$.

• For $x$, we have $x^2 - 2x$. Completing the square: $x^2 - 2x = (x - 1)^2 - 1$

• For $y$, we have $y^2 + 6y$. Completing the square: $y^2 + 6y = (y + 3)^2 - 9$

Now, rewrite the equation with these squared terms: $(x - 1)^2 - 1 + (y + 3)^2 - 9 = 26$ Simplify: $(x - 1)^2 + (y + 3)^2 = 26 + 1 + 9$ $(x - 1)^2 + (y + 3)^2 = 36$

Step 2: Finding the center and radius

• The standard form of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Comparing this with $(x - 1)^2 + (y + 3)^2 = 36$, we can see that:
  • The center of the circle is $(1, -3)$.
  • The radius $r$ is $\sqrt{36} = 6$.

Answer:

• The coordinates of the center are $(1, -3)$.
• The radius is $6$.

Question 2

The circle $C$ has center $(2, 5)$ and passes through the point $(4, 9)$. Find the equation for $C$.

Step 1: Use the formula of a circle

The standard equation of a circle is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

Step 2: Finding the radius

We are given the center $(2, 5)$ and a point on the circle $(4, 9)$. The radius is the distance between these two points: $r = \sqrt{(4 - 2)^2 + (9 - 5)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

Step 3: Writing the equation of the circle

The equation of the circle is: $(x - 2)^2 + (y - 5)^2 = (2\sqrt{5})^2$ $(x - 2)^2 + (y - 5)^2 = 20$

Answer: The equation of the circle is $(x - 2)^2 + (y - 5)^2 = 20$.

Question 3

The circle $C$ has center $(-2, 3)$ and passes through point $(1, 8)$.

1. (a) Find the equation for C.

Step 1: Use the formula of a circle

The center is $(-2, 3)$ and the radius can be found by calculating the distance from $(-2, 3)$ to $(1, 8)$: $r = \sqrt{(1 - (-2))^2 + (8 - 3)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}$

Step 2: Writing the equation of the circle

The equation of the circle is: $(x + 2)^2 + (y - 3)^2 = (\sqrt{34})^2$ $(x + 2)^2 + (y - 3)^2 = 34$

Answer: The equation of the circle is $(x + 2)^2 + (y - 3)^2 = 34$.

2. (b) Show that the point $(3, 6)$ lies on C.

Substitute $x = 3$ and $y = 6$ into the equation: $(3 + 2)^2 + (6 - 3)^2 = 34$ $5^2 + 3^2 = 25 + 9 = 34$ Since both sides are equal, the point $(3, 6)$ lies on the circle.

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