Math / Geometry

QuestionWrite the standard form of the equation of the circle with the given center and radius. Center $(-4,5), r=\sqrt{5}$
The equation of the circle in standard form is $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a circle, given its center and radius, and write it in a standard format. Watch out! Don't mix up the signs of the coordinates of the center when plugging them into the equation!
Also, remember to square the radius correctly.

STEP 2

1. Recall the standard equation
2. Plug in the values
3. Simplify the equation

STEP 3

Alright, let's kick things off!
The standard equation of a circle is given by $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center of the circle, and $r$ is its radius.
This equation tells us that the distance squared between any point $(x, y)$ on the circle and the center $(h, k)$ is always equal to the radius squared.
Pretty neat, huh?

STEP 4

We're given that the center is at $(-4, 5)$ , so $h = -4$ and $k = 5$ .
The radius is $r = \sqrt{5}$ .
Let's plug these values into our standard equation: $(x - (-4))^2 + (y - 5)^2 = (\sqrt{5})^2$

STEP 5

Time to simplify!
Notice the double negative in the first term.
Subtracting a negative is the same as adding, so $(x - (-4))$ becomes $(x + 4)$ .
Let's rewrite the equation: $(x + 4)^2 + (y - 5)^2 = (\sqrt{5})^2$

STEP 6

Now, let's simplify the right side.
Squaring a square root removes the root, so $(\sqrt{5})^2$ becomes 5.
Our equation now looks like this: $(x + 4)^2 + (y - 5)^2 = 5$ This is the standard form of the equation of the circle!

STEP 7

The standard form of the equation of the circle is $(x + 4)^2 + (y - 5)^2 = 5$ .

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QuestionWrite the standard form of the equation of the circle with the given center and radius. Center $(-4,5), r=\sqrt{5}$
The equation of the circle in standard form is $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a circle, given its center and radius, and write it in a standard format. Watch out! Don't mix up the signs of the coordinates of the center when plugging them into the equation!
Also, remember to square the radius correctly.

STEP 2

1. Recall the standard equation
2. Plug in the values
3. Simplify the equation

STEP 3

STEP 4

We're given that the center is at $(-4, 5)$ , so $h = -4$ and $k = 5$ .
The radius is $r = \sqrt{5}$ .
Let's plug these values into our standard equation: $(x - (-4))^2 + (y - 5)^2 = (\sqrt{5})^2$

STEP 5

Time to simplify!
Notice the double negative in the first term.
Subtracting a negative is the same as adding, so $(x - (-4))$ becomes $(x + 4)$ .
Let's rewrite the equation: $(x + 4)^2 + (y - 5)^2 = (\sqrt{5})^2$

STEP 6

Now, let's simplify the right side.
Squaring a square root removes the root, so $(\sqrt{5})^2$ becomes 5.
Our equation now looks like this: $(x + 4)^2 + (y - 5)^2 = 5$ This is the standard form of the equation of the circle!

STEP 7

The standard form of the equation of the circle is $(x + 4)^2 + (y - 5)^2 = 5$ .

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QuestionWrite the standard form of the equation of the circle with the given center and radius. Center (−4,5),r=5(-4,5), r=\sqrt{5}(−4,5),r=5​The equation of the circle in standard form is □\square□ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a circle, given its center and radius, and write it in a standard format. Watch out! Don't mix up the signs of the coordinates of the center when plugging them into the equation!Also, remember to square the radius correctly.

STEP 2

1. Recall the standard equation2. Plug in the values3. Simplify the equation

STEP 3

STEP 4

We're given that the **center** is at (−4,5)(-4, 5)(−4,5), so h=−4h = -4h=−4 and k=5k = 5k=5.The **radius** is r=5r = \sqrt{5}r=5​.Let's plug these values into our **standard equation**: (x−(−4))2+(y−5)2=(5)2(x - (-4))^2 + (y - 5)^2 = (\sqrt{5})^2(x−(−4))2+(y−5)2=(5​)2

STEP 5

Time to simplify!Notice the double negative in the first term.Subtracting a negative is the same as adding, so (x−(−4))(x - (-4))(x−(−4)) becomes (x+4)(x + 4)(x+4).Let's rewrite the equation: (x+4)2+(y−5)2=(5)2(x + 4)^2 + (y - 5)^2 = (\sqrt{5})^2(x+4)2+(y−5)2=(5​)2

STEP 6

Now, let's simplify the right side.Squaring a square root removes the root, so (5)2(\sqrt{5})^2(5​)2 becomes **5**.Our equation now looks like this: (x+4)2+(y−5)2=5(x + 4)^2 + (y - 5)^2 = 5(x+4)2+(y−5)2=5 This is the **standard form** of the equation of the circle!

STEP 7

The standard form of the equation of the circle is (x+4)2+(y−5)2=5(x + 4)^2 + (y - 5)^2 = 5(x+4)2+(y−5)2=5.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionWrite the standard form of the equation of the circle with the given center and radius. Center $(-4,5), r=\sqrt{5}$
The equation of the circle in standard form is $\square$ (Simplify your answer.)

What is this asking? We need to find the equation of a circle, given its center and radius, and write it in a standard format. Watch out! Don't mix up the signs of the coordinates of the center when plugging them into the equation!
Also, remember to square the radius correctly.

1. Recall the standard equation
2. Plug in the values
3. Simplify the equation

We're given that the center is at $(-4, 5)$ , so $h = -4$ and $k = 5$ .
The radius is $r = \sqrt{5}$ .
Let's plug these values into our standard equation: $(x - (-4))^2 + (y - 5)^2 = (\sqrt{5})^2$

Time to simplify!
Notice the double negative in the first term.
Subtracting a negative is the same as adding, so $(x - (-4))$ becomes $(x + 4)$ .
Let's rewrite the equation: $(x + 4)^2 + (y - 5)^2 = (\sqrt{5})^2$

Now, let's simplify the right side.
Squaring a square root removes the root, so $(\sqrt{5})^2$ becomes 5.
Our equation now looks like this: $(x + 4)^2 + (y - 5)^2 = 5$ This is the standard form of the equation of the circle!

The standard form of the equation of the circle is $(x + 4)^2 + (y - 5)^2 = 5$ .