Question

If the relation R is the set A of points, in a plane given by $R=\left\{ \left( P,Q \right) \right\}\,$distance of the point P from the origin is same as the distance of the point Q from the origin. is an equivalence relation, then the set of all points related to a point $P\ne \left( 0,0 \right)$ is the?
(a) Circle
(b) Ellipse
(c) Parabola
(d) Hyperbola

Answer 1

Hint: In this question, we will check the condition of circle, ellipse, parabola and hyperbola with condition of relation R to find the correct option.

Complete step-by-step solution -
As we know, a circle is that two- dimensional geometrical figure, whose all the points on the boundary are equidistant from one fixed point, which is the centre of the circle.
Ellipse is that two-dimensional figure; whose all the points of the curve have a fixed sum of distance from two fixed points, which are known as foci of the ellipse.
Parabola is that two-dimensional figure, whose all the points on the curve have equal distance from one fixed point, known as focus of parabola and one fixed line, known as direct nix of parabola.
Hyperbola is that two-dimensional figure, whose all the points on the curve have equal difference from two fixed points, known as foci of parabola.
Now, in a given question, we have set A of all points in a plane.
We have a relation R defined on this set A such that $R=\left\{ \left( P,Q \right) \right\}\,$distance of the point P from the origin is the same as the distance of the point Q from the origin.
So, the focus of all points in the plane which belongs to R will be all the points which are equidistant from one fixed point, and $P\ne \left( 0,0 \right)$.
Then from above definitions, we see that this is the condition of the circle, where one fixed point is the centre of the circle. Since $P\ne \left( 0,0 \right)$, the centre of this circle will be $\left( 0,0 \right)$.
Hence, the correct answer is option (a).

Note: In this type of question, we are not to consider finite number of points R but instead locus of points of R, which will be infinite.