The explicit rule is given. Write the recursive rule. Show your work. an=30 n−84

The locus of the midpoints of all chords that can be drawn from a given fixed point 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 and 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 According to the midpoint formula, the coordinates of the midpoint of the line segment joining the points and is

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 , that is,

Let the coordinates of the given fixed point be

Let the coordinates of any point on the circle be and let the coordinates of the midpoint of the chord joining the points and be

We have to find the locus of

Then, using the midpoint formula,

On solving, we get,

Since and are both points on the given circle, they satisfy the equation of the circle,

Then,

Substituting in , we have,

This is the locus of the point

Replace to get,

This is the equation of a circle with center at 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 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 and respectively.

Learn more about locus here: