Use the grouping method to factor x^3+x^2+3x+3

The circle Ci, intersects the y-axis at two points, one of which is (0.4).

Anuta_ua [19.1K]

Answer:

Part 1) r=5 units (see the explanation)

Part 2) $(x-4)^2+(y-7)^2=25$

Part 3) The center of the circle is (-3,4) and the radius is 4 units

Part 4) see the explanation

Step-by-step explanation:

Part 1)

step 1

Find the center of circle C_1

we know that

The distance between the center and point (0,4) is equal to the radius

The distance between the center and point (4,2) is equal to the radius

Let

(x,y) ----> the coordinates of center of the circle

Remember that

The tangent y=2 (horizontal line) to the circle is perpendicular to the radius of the circle at point (4,2)

That means ----> The segment perpendicular to the tangent is a vertical line x=4

so

The x-coordinate of the center is x=4

The coordinates of center are (4,y)

the formula to calculate the distance between two points is equal to

$d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}$

Remember

The distance between the center (4,y) and point (0,4) is equal to the radius

The distance between the center (4,y) and point (4,2) is equal to the radius

so

substitute

$\sqrt{(y-4)^{2}+(4-0)^{2}}=\sqrt{(4-4)^{2}+(y-2)^{2}}$

$\sqrt{(y-4)^{2}+16}=\sqrt{(0)^{2}+(y-2)^{2}}$

squared both sides

$(y-4)^{2}+16=(y-2)^{2}$

solve for y

$y^2-8y+16+16=y^2-4y+4$

$y^2-8y+32=y^2-4y+4\\8y-4y=32-4\\4y=28\\y=7$

The coordinates of the center are (4,7)

step 2

Find the radius of circle C_1

$r=\sqrt{(y-4)^{2}+(4-0)^{2}}$

substitute the value of y

$r=\sqrt{(7-4)^{2}+(4-0)^{2}}$

$r=\sqrt{(3)^{2}+(4)^{2}}$

$r=\sqrt{25}$

$r=5\ units$

Part 2)

Find the equation of the circle C, in standard form.

we know that

The equation of a circle in standard form is

$(x-h)^2+(y-k)^2=r^2$

where

(h,k) is the center

r is the radius

substitute the given values

$(x-4)^2+(y-7)^2=5^2$

$(x-4)^2+(y-7)^2=25$

Part 3) Another circle C2 has equation x² + y2 + 6x – 8y +9=0

Find the centre and radius of C2

we have

$x^2+y^2+6x-8y+9=0$

Convert to standard form

$(x-h)^2+(y-k)^2=r^2$

where

(h,k) is the center

r is the radius

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$(x^2+6x)+(y^2-8y)=-9$

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

$(x^2+6x+9)+(y^2-8y+16)=-9+9+16$

$(x^2+6x+9)+(y^2-8y+16)=16$

Rewrite as perfect squares

$(x+3)^2+(y-4)^2=16$

$(x+3)^2+(y-4)^2=4^2$

therefore

The center of the circle is (-3,4) and the radius is 4 units

Part 4) Show that the circle C2 is a tangent to the x-axis

we know that

If the x-axis is tangent to the circle, then the equation of the tangent is y=0

so

The radius of the circle must be perpendicular to the tangent

That means ----> The segment perpendicular to the tangent is a vertical line The equation of the vertical line is equal to the x-coordinate of the center

so

x=-3

The circle C_2, intersects the x-axis at point (-3,0)

<em>Verify</em>

The distance between the center (-3,4) and point (-3,0) must be equal to the radius

Calculate the radius

$r=\sqrt{(0-4)^{2}+(-3+3)^{2}}$

$r=\sqrt{16}$

$r=4\ units$ ----> is correct

therefore

The circle C_2 is tangent to the x-axis

7 0

3 years ago

Read 2 more answers

PLEASE I NEED HELP! I will give a good rating!

Phantasy [73]

Answer:

paralelogram.

Step-by-step explanation:

Make a graph, plot the points and draw lines connecting them. Make your own conclusion because I cant write this for you. :)

I would use length for this

7 0

3 years ago

what is the volume of the largest sphere that you could carve out of a wooden block whose edges measure 8 m by 8 m by 8 m? Use t

Papessa [141]

Given data:

The given cube .

The expression for the volume of the sphere is,

$\begin{gathered} V=\frac{4}{3}\pi(\frac{8}{2}m)^{3^{}} \\ =268.0826m^3 \\ \approx261m^3 \end{gathered}$

Thus, the volume of the sphere is 261 cubic-m.

7 0

1 year ago

19) A six-sided die is rolled six times. What is

aleksley [76]

Answer:

D

Step-by-step explanation:

Probability that the die will show an even number

in one rolling is 3/6=0,5 = p

We have formula for repeating rolling

n!/(r!(n-r)!) p^r (1-p)^(n-r)

n = how many times we repeat rolling = 6

r = how many times we want something to happen

(show an even number exactly two times) = 2

15 *(0,5)^2 *(0,5)^4 = 0,234375

4 0

2 years ago

3/5 × -4/7 + 4/35 - 3/10 × 4/7

Annette [7]

Answer:

I think its -2/5 but not sure

Step-by-step explanation:

5 0

3 years ago

Read 2 more answers