Mathematics, 27.07.2021 21:40 rebekahlm

Please Help! What is the locus of the midpoints of all chords that can be drawn from a given point on a circle with a radius of 6.

Answers: 3

Show answers

Answers

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

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:

Answer from: Quest

the answer is 28

step-by-step explanation:

Answer from: Quest

Element 17 is chlorine it is a group 7 element(halogen) and belongs to period 3 of the periodic table

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 22:30, jaydenrenee111902

The volume of the sun is approximately 1 × 10^18 cu km, while the volume of jupiter is approximately 1 × 10^15 cu km. which of the following is true? a. the volume of jupiter is approximately one thousand times the volume of the sun. b. the volume of jupiter is approximately one hundred times the volume of the sun. c. the volume of the sun is approximately one thousand times the volume of jupiter. d. the volume of the sun is approximately one hundred times the volume of jupiter.

Answers: 2

continue

Mathematics, 22.06.2019 02:40, carms12

There are 18 female students and 16 male students in a class. which of the following expresses the ratio of female students to male students in simplest form? а) 10 b) 9 to 8 c) 8: 9 d)18/16

Answers: 1

continue

Mathematics, 22.06.2019 03:00, crystalbyrd79p8imrx

What is the slopeof the live that passes through the points and (-2,-5) and (3,-1)

Answers: 2

continue

Mathematics, 22.06.2019 03:30, coolgirl5679

The half-life of a certain material is 0.002 second. approximately how much of a 14-gram sample will be left after 0.004 second? a. 12 grams b. 3.5 grams c. 0.06 gram d. 7 grams

Answers: 3

continue

You know the right answer?

Please Help! What is the locus of the midpoints of all chords that can be drawn from a given point...