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This is NOT true.
The cardinality of the set of points in a circle is the same as the cardinality of the set of points in a line.
First, break the circle and straighten it out. I think you would agree that the number of points remains the same.
Now apply some continuous monotonic function that takes one end of that line segment and assigns it to -infinity and the other end to +infinity. I think you would agree that this is possible.
We have now made a one-to-one, invertible correspondence between the points in the original circle and the points in a line, demonstrating that the two objects have the same cardinality.
Roughly speaking!
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What is a straight line joining points on a circle?
A straight line joining points on a circle is called a "chord" of that circle. If the line happens to pass through the center of the circle, then it's a "diameter" of that circle. The question asked about "points" on a circle, so two points on the circumference of that circle are being considered. (No line can join more than two points of a circle.)
Can a straight line intersect a circle at more than two points?
No.
How can you prove that the line between to points is a tangent of the smaller circle?
Step I: Show that both points are outside the smaller circles. Possibly by showing that distance from each point to the centre of the circle is greater than its radius. Step 2: Show that the line between the two points touches the circle at exactly one point. This would be by simultaneous solution of the equations of the line and the circle.
Which point does not lie on the circle centered at A(3 1) and passing through the origin (0 0)?
Lots of points don't lie on the circle. In fact, there are (in a way) more points NOT on the circle, than points on the circle.
What is the locus of points less than three units from a given point?
All points less than 3 distant from point 'P' comprise a circle, centered at 'P', with a radius of 3, but NOT including the line that is the circumference of the circle.
