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
Phantasy [73]
3 years ago
8

Please Help!

Mathematics
1 answer:
oksano4ka [1.4K]3 years ago
5 0

The locus of the midpoints of all chords that can be drawn from a given fixed point (a,b) on a circle with a radius of 6 units, is a circle of radius 3 units with center at a point whose x & y coordinates are shifted from the center of the given circle by \frac{a}{2} and \frac{b}{2} respectively.

Given: A circle of radius 6 units

To find: The locus of the midpoint of all chords that can be drawn from a given point on the circle.

To find the required locus, we need to know the following:

  • Locus of a moving point is the trajectory of that point. It is the geometrical figure represented by the equation which is satisfied by the coordinates of the moving point.
  • A chord of a circle is a line segment joining any points of a circle.
  • Equation of a circle with center at origin and radius of r units is x^{2} +y^{2} =r^{2}
  • According to the midpoint formula, the coordinates of the midpoint of the line segment joining the points (x_{1},y_{1}) and (x_{2},y_{2}) is (\frac{x_{1}+x_{2} }{2} ,\frac{y_{1}+y_{2} }{2} )

Let, without loss of generality, the given circle be centered at the origin. Even if it is not, we can shift the origin to the center of the given circle with coordinate transformation.

Then, the equation of the given circle is x^{2}+y^{2} =6^{2}, that is, x^{2}+y^{2} = 36

Let the coordinates of the given fixed point be (a,b)

Let the coordinates of any point on the circle be (p,q) and let the coordinates of the midpoint of the chord joining the points (a,b) and (p,q) be (h,k)

We have to find the locus of (h,k)

Then, using the midpoint formula,

(h,k)=(\frac{a+p}{2} ,\frac{b+q}{2})

On solving, we get,

p=2h-a,q=2k-b

Since (a,b) and (p,q) are both points on the given circle, they satisfy the equation of the circle, x^{2}+y^{2} = 36

Then,

a^{2} +b^{2} =36

p^{2} +q^{2} =36

Substituting p=2h-a,q=2k-b in p^{2} +q^{2} =36, we have,

(2h-a)^{2} +(2k-b)^{2} =36

(2(h-\frac{a}{2}) )^{2} +(2(k-\frac{b}{2}))^{2} =36

4(h-\frac{a}{2})^{2} +4(k-\frac{b}{2})^{2} =36

(h-\frac{a}{2})^{2} +(k-\frac{b}{2})^{2} =\frac{36}{4}

(h-\frac{a}{2})^{2} +(k-\frac{b}{2})^{2} =9

(h-\frac{a}{2})^{2} +(k-\frac{b}{2})^{2} =3^{2}

This is the locus of the point (h,k)

Replace (h,k)=(x,y) to get,

(x-\frac{a}{2})^{2} +(y-\frac{b}{2})^{2} =3^{2}

This is the equation of a circle with center at (\frac{a}{2} ,\frac{b}{2} ) and radius 3 units.

Thus, we can conclude that the locus of the midpoints of all chords that can be drawn from a given fixed point (a,b) on a circle with a radius of 6 units, is a circle of radius 3 units with center at a point whose x & y coordinates are shifted from the center of the given circle by \frac{a}{2} and \frac{b}{2} respectively.

Learn more about locus here:

brainly.com/question/23824483

You might be interested in
Which equation represents the graphed function?
uysha [10]

Answer:

Step-by-step explanation:

6x

6 0
3 years ago
Tori went to see a monster truck show at the local arena before the show Tori got to see the truck up close he noticed that each
AlexFokin [52]

Answer:

37.70 feet

Step-by-step explanation:

-The radius of each tire is given as 6 feet.

-Distance around each tire is equivalent to the circumference of each tire.

#Circumference is calculated as:

C=\pi D=2\pi r\\\\=2\pi \times 6 \\\\=12\pi\\\\=37.70\ feet

Hence, the distance around each tire is 37.70 feet

7 0
3 years ago
PLZ HELP DUE SOON!!!!!!
ANEK [815]
Let's factor the following polynomial to find the roots:
 x2-13x + q = 0
 Factoring we have:
 (x-12.5) * (x + a)
 Where,
 -12.5 + a = -13
 a = -13 + 12.5
 a = -0.5
 We now look for the coefficient q:
 q = -12.5 * a
 q = -12.5 * (- 0.5)
 q = 6.25
 The polynomial will then be:
 x2-13x + 6.25 = 0
 Its roots are:
 x1 = 12.5
 x2 = 0.5
 Answer:
 The other root and the value of the coefficient are:
 0.5 and 6.25 (Respectively)
3 0
3 years ago
Is 2x+4-x linear or non linear?
lawyer [7]

Answer:

To determine if an equation is a linear function, it must have the form y = mx + b (in which m is the slope and b is the y-intercept). A nonlinear function will not match this form.

So it is linear.

6 0
2 years ago
The set of x-values for a relation is also called the ? ( domaing or range )
sukhopar [10]

the set of x values are called the domain and the corresponding values of y the range

7 0
3 years ago
Other questions:
  • workers on the simile line produce 4 x + 6 bolts each day which expression shows how many books they produced in 12 days
    12·1 answer
  • Find the volume of a square pyramid with a base area of 18inch squared and the height of 4 inches
    12·1 answer
  • How many 4-letter passords can be made using the letters A thought Z if...
    8·1 answer
  • What is the square root of 9​
    9·2 answers
  • A surveyor is 40m from the edge of a building. The angle of elevation from the surveyor to the top of the building is 55° . What
    8·1 answer
  • Points A, B, C, and D lie on the circle. Solve for the value of X
    11·2 answers
  • i really really really need help with this math question i’m giving away FREE points and BRAINLIEST for correct answer
    7·2 answers
  • How to do this question plz answer me ​
    5·1 answer
  • If you have a deck of cards what is the probability that you will draw a 9 or a heart
    5·1 answer
  • Cube M has an edge length of
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!