1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
S_A_V [24]
2 years ago
15

Find the equation of the circle having radius 5 and which touches the line 3x+4y-16=0 at point (4,1)?

Mathematics
1 answer:
Sladkaya [172]2 years ago
5 0

The only information missing is the circle's center; let it be the point (a, b). Then the equation of the circle is

(x - a)² + (y - b)² = 5²

We know the point (4, 1) lies on the circle, so we have the condition

(4 - a)² + (1 - b)² = 5²   ⇒   (a - 4)² + (b - 1)² = 25

The circle just touches the line, so the line is tangent to the circle. The slope of the line is -3/4, since

3x + 4y - 16 = 0   ⇒   y = 4 - 3/4 x

Any non-vertical tangent line to the circle at (a', b') has slope equal to dy/dx with x = a' and y = b'. Differentiating both sides of the circle equation with respect to x yields

2 (x - a) + 2 (y - b) dy/dx = 0   ⇒   dy/dx = -(x - a)/(y - b)

At the point (4, 1), the slope of the tangent is -3/4, so we get the condition

-(4 - a)/(1 - b) = -3/4   ⇒   4a - 3b = 13

Solve for b in terms of a and substitute into the other condition to solve for a :

4a - 3b = 13   ⇒   b = (4a - 13)/3

(a - 4)² + ((4a - 13)/3 - 1)² = 25

25/9 a² - 200/9 a + 400/9 = 25

25/9 a² - 200/9 a + 175/9 = 0

a² - 8a + 7 = 0

(a - 7) (a - 1) = 0

a = 7   or   a = 1

Solve for b :

b = (4a - 13)/3   ⇒   b = 5   or   b = -3

So there are two possible circles,

(x - 7)² + (y - 5)² = 5²

or

(x - 1)² + (y + 3)² = 5²

You might be interested in
The ratio of boys to girls in the ninth grade is 6:5. If there are 185 girls in the class, how many boys are there.
FrozenT [24]
There are 222 boys in the class. You find this by dividing 185 by 5 which gets you 37, so you have to multiply the 6 by 37 to find the amount of boys.
4 0
3 years ago
Read 2 more answers
Equation of the line that passes through the points (-3,8) and has a slope of zero
stiks02 [169]

Answer:

no

Step-by-step explanation:

6 0
3 years ago
What is the slope of a line that passes (4, 10) and (1, 10)
SpyIntel [72]

Answer:

The slope of the a straight line is given by the ratio of the Rise to the Run

of the line. The rise between the given points is zero.

The slope of the line that passes through the points (4, 10) and (1, 10) is zero.

Step-by-step explanation:

The given points are; (4, 10) and (1, 10)

The slope of a line, m, is given by the following formula;

Where;

(x₁, y₁) = (4, 10) and (x₂. y₂) = (1, 10), we get;

The slope of the line that passes through the points (4, 10) and (1, 10) is 0.

3 0
2 years ago
Read 2 more answers
Please help me with this
Soloha48 [4]

for an isosceles right triangle the legs would be 2 times the square root of the hypotenuse

 so for this:

the legs would be 2sqrt(10)


3 0
3 years ago
What is 11- 9 1/4 simplfy
bija089 [108]

1 and 3/4 is the answer!

Hope this helps! May I have brainliest? :D

8 0
2 years ago
Other questions:
  • A pet store has 10 puppies for every 6 kittens. What is the ratio of kittens to puppies?
    14·1 answer
  • If Mei has earned 30 points right and solve a multiplication equation to find out how many books she needs to read
    13·2 answers
  • Suppose a hot air balloon begins descending at a rate of 30ft/min from an elevation of 1250ft. Write an equation to model the ho
    15·1 answer
  • The sum of 2 consecutive integers divided by 4 is 189.5
    12·1 answer
  • Find the difference and simplify your answer correctly<br> 7/12-2/5
    15·1 answer
  • rewarding brainly for correct answer and if you dont get brainly lmk if you want me to make a post to give it to you or try and
    8·2 answers
  • A line passes through (-1,-5) and is perpendicular to the graph of y = -2x + 5. What equation represents the line in
    13·1 answer
  • Solve for x. 42-3x=30.<br> Please and thank you!<br> will mark brainliest!
    6·2 answers
  • Haiiiii i need help with this question pls!! what is the distance between (3,3) and (-5,-2)
    11·1 answer
  • Evaluate squareroot of <br> 61.73²-38.27²
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!