Please help me! This is a quiz!

For which value(s) of the constant k is the circle x² + (y − k)² = 16 tangent to the line y = 3?

STALIN [3.7K]

Answer:

Step-by-step explanation:Let us find points of intersection of line

3

x

+

4

y

−

k

=

0

and circle

x

2

+

y

2

=

16

. We can do this by putting value of

y

from first equation i.e.

y

=

k

−

3

x

4

and we get

x

2

+

(

k

−

3

x

)

2

16

=

16

or

16

x

2

+

k

2

+

9

x

2

−

6

k

x

=

256

i.e.

25

x

2

−

6

k

x

+

k

2

−

256

=

0

This would give two values of

x

and corresponding two values of

y

i.e. two points. But tangent cuts the circle in only at one point. This will be so when discriminant is zero i.e.

(

−

6

k

)

2

−

4

⋅

25

⋅

(

k

2

−

256

)

=

0

or

−

64

k

2

+

25600

=

0

or

k

=

±

20

graph{(x^2+y^2-16)(3x+4y-20)(3x+4y+20)=0 [-10, 10, -5, 5]}

5 0

4 years ago

Read 2 more answers

Enlarge the triangle by scale factor of -1/3

Bess [88]

The attached image represents the triangle after it has been enlarged

<h3>How to enlarge the triangle?</h3>

The given parameters are:

Scale factor, k = -1/3
Center, (a,b) = (-1,2)

From the graph, the coordinates of the triangle are:

A = (-4,2)

B = (-4,-4)

C = (-1,-4)

The rule of dilation is calculated using:

$(x,y) \to (k(x - a) + a, k(y - b) + b)$

So, we have:

$A' = (-\frac 13(-4 +1) - 1, -\frac 13(2 - 2) + 2)$

A' = (0, 2)

$B' = (-\frac 13(-4 +1) - 1, -\frac 13(-4 - 2) + 2)$

B' = (0, 4)

$C' = (-\frac 13(-1 +1) - 1, -\frac 13(-4 - 2) + 2)$

C' = (-1, 4)

See attachment for the image of the triangle, after it has been enlarged

Read more about dilation at:

brainly.com/question/3457976

#SPJ1

4 0

2 years ago

Let |u| = 10 at an angle of 45° and |v| = 13 at an angle of 150°, and w = u + v. What is the magnitude and direction angle of w?

mixer [17]

Given:

$|u|=10$ at an angle of 45°.

$|v|=13$ at an angle of 150°.

$w=u+v$

To find:

Magnitude and direction angle of w.

Solution:

We have,

$w=u+v$

$\theta = 45^\circ, \phi = 150^\circ$

Now,

$u_x=|u|\cos \theta=10\cos 45^\circ=7.071$

$u_y=|u|\sin \theta=u_y=10\sin 45^\circ=7.071$

$v_x=|v|\cos \phi=13\cos 150^\circ=-11.25833$

$v_y=|v|\sin \phi=u_y=13\sin 150^\circ=6.5$

Using these information, we get

$R_x=u_x+v_x=7.071-11.25833=-4.18733$

$R_y=u_y+v_y=7.071+6.5=13.571$

$\text{Direction angle}=\tan^{-1}(\dfrac{R_y}{R_x})$

$\text{Direction angle}=\tan^{-1}(\dfrac{13.571}{-4.18733})$

$\text{Direction angle}=-72.8239$

$\text{Direction angle}=107.1476$

$\text{Direction angle}\approx 107.1$

Now,

$|w|=\sqrt{(|u|\cos \theta +|v|\cos \phi)^2+(|u|\sin \theta +|v|\sin \phi)^2}$

$|w|=\sqrt{(10\cos(45)+13\cos 150)^2+(10\sin(45)+13\sin 150)^2}$

$|w|=\sqrt{(10(\dfrac{1}{\sqrt{2}})+13(-\dfrac{\sqrt{3}}{2}))^2+(10(\dfrac{1}{\sqrt{2}})+13(\dfrac{1}{2}))^2}$

$|w|=14.20236$

$|w|\approx 14.20236$

Therefore, the correct option is D.

4 0

3 years ago

Read 2 more answers

A man realizes he lost the detailed receipt from the store and only has the credit card receipt with the after-tax total. If the

Eva8 [605]

Heyyy hit me up if u want help.

3 0

3 years ago

Read 2 more answers

Consider the following: A medical researcher wants to determine the effects of pH (a measure of alkalinity or acidity, with pure

natita [175]

Answer:

See below

Step-by-step explanation:

<h3>Continuation of the question</h3>

Data in the table:

PH 4 5 6 7 8 9 10 11

Bacteria 116 120 131 136 141 151 148 163

Create a scatter plot to represent the data: Based on the data and your scatter plot, what do you think a good linear function would be to represent the data?

<h3>Solution</h3>

See attached for scatter plot

We can see the chart is close to linear function. Let's build one based on data in the table

The line will be in the form of:

y = mx + b

If we use two points and get the equation

(4, 116) and (8, 141)

Finding the slope:

m = (141-116)/(8-4) = 25/4 = 6.25

Then, finding y-intercept:

116 = 6.25*4 + b
b = 116 - 25 = 91

So the line is:

y = 6.25x + 91

Note: This is approximate best fit graph and it is attached as well. Exact formula for the line is built by using different method and is very close to the one above

4 0

4 years ago