Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$r:$ $A$ circle is a particular case of an ellipse.
$r:$ $A$ circle is a particular case of an ellipse.
(A) The standard equation of an ellipse is given by $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$.
If we set $a = b = r$,the equation becomes $\frac{x^{2}}{r^{2}} + \frac{y^{2}}{r^{2}} = 1$,which simplifies to $x^{2} + y^{2} = r^{2}$.
This is the standard equation of a circle with radius $r$ centered at the origin.
Since a circle can be derived from an ellipse by setting the semi-major and semi-minor axes equal,a circle is indeed a particular case of an ellipse.
Therefore,statement $r$ is true.
If we set $a = b = r$,the equation becomes $\frac{x^{2}}{r^{2}} + \frac{y^{2}}{r^{2}} = 1$,which simplifies to $x^{2} + y^{2} = r^{2}$.
This is the standard equation of a circle with radius $r$ centered at the origin.
Since a circle can be derived from an ellipse by setting the semi-major and semi-minor axes equal,a circle is indeed a particular case of an ellipse.
Therefore,statement $r$ is true.
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