(C) If an equation $f(x, y) = 0$ remains unchanged when $x$ is replaced by $-x$ and $y$ is replaced by $-y$,then $f(-x, -y) = f(x, y) = 0$. This condi

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Symmetric in opposite quadrants

(C) If an equation $f(x, y) = 0$ remains unchanged when $x$ is replaced by $-x$ and $y$ is replaced by $-y$,then $f(-x, -y) = f(x, y) = 0$. This condition implies that for every point $(x, y)$ on the curve,the point $(-x, -y)$ also lies on the curve. This geometric property is known as symmetry about the origin or symmetry in opposite quadrants.

Class 11 Mathematics 10-1.Circle and System of Circles Locus Related Problem

Medium

If the equation of a curve remains unchanged by replacing $x$ and $y$ with $-x$ and $-y$ respectively,then the curve is

A
Symmetric along the $x$-axis
B
Symmetric along the $y$-axis
C
Symmetric in opposite quadrants
D
Symmetric along the line $y = x$

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10-1.Circle and System of Circles

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Locus Related Problem

$A$ variable circle passes through the fixed point $(2,0)$ and touches the $y$-axis. Then the locus of its centre is

Medium

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The locus of the mid-points of the chords of the circle $x^{2}+y^{2}+2x-2y-2=0$ which make an angle of $90^{\circ}$ at the centre is

DifficultWBJEE 2017

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The locus of the centre of a circle touching both the coordinate axes is

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$A$ circle passes through the point $(3, 4)$ and cuts the circle $x^2 + y^2 = a^2$ orthogonally. The locus of its centre is a straight line. If the distance of this straight line from the origin is $25$,then $a^2$ is equal to:

MediumAP EAMCET 2012

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For $t \in (0, 2\pi)$,if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$,$B(\cos t, \sin t)$,and $C(a, b)$ such that its orthocentre lies on a circle with centre $(1, 1/3)$,then $(a^2 - b^2)$ is equal to.

DifficultJEE MAIN 2022

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If the equation of a curve remains unchanged by replacing $x$ and $y$ with $-x$ and $-y$ respectively,then the curve is

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$A$ variable circle passes through the fixed point $(2,0)$ and touches the $y$-axis. Then the locus of its centre is

The locus of the mid-points of the chords of the circle $x^{2}+y^{2}+2x-2y-2=0$ which make an angle of $90^{\circ}$ at the centre is

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$A$ circle passes through the point $(3, 4)$ and cuts the circle $x^2 + y^2 = a^2$ orthogonally. The locus of its centre is a straight line. If the distance of this straight line from the origin is $25$,then $a^2$ is equal to:

For $t \in (0, 2\pi)$,if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$,$B(\cos t, \sin t)$,and $C(a, b)$ such that its orthocentre lies on a circle with centre $(1, 1/3)$,then $(a^2 - b^2)$ is equal to.

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