Understanding the Problem
To find the equation of a circle that passes through the origin $(0,0)$ and makes equal intercepts on the coordinate axes, we must first recognize the general form of a circle equation:
$x^2 + y^2 + 2gx + 2fy + c = 0$
Given the conditions:
- Passes through the origin: If $(0,0)$ lies on the circle, substituting $x=0$ and $y=0$ into the equation gives $0 + 0 + 0 + 0 + c = 0$, so $c = 0$.
- Equal intercepts on axes: A circle cuts an x-intercept of length $2\sqrt{g^2 - c}$ and a y-intercept of length $2\sqrt{f^2 - c}$. Since $c=0$, these simplify to $2|g|$ and $2|f|$. For the intercepts to be equal, we set $2|g| = 2|f|$, which implies $|g| = |f|$. This leads to two cases: $g = f$ or $g = -f$.
Step-by-Step Derivation
Since the circle passes through the origin and $c=0$, the equation becomes: $x^2 + y^2 + 2gx + 2fy = 0$
Let the x-intercept be $a$ and the y-intercept be $a$. The x-intercept is found by setting $y=0$ in the equation: $x^2 + 2gx = 0 \Rightarrow x(x + 2g) = 0$ So the intercepts are at $(0,0)$ and $(-2g, 0)$. Thus, the length of the x-intercept is $|-2g| = |a|$. Similarly, for the y-intercept, setting $x=0$ gives: $y^2 + 2fy = 0 \Rightarrow y(y + 2f) = 0$ The intercepts are at $(0,0)$ and $(0, -2f)$. The length is $|-2f| = |a|$.
Given that the intercepts are equal in magnitude and sign (assuming the intercept length $a$ is non-zero), we can define the intercept as $a = -2g$ and $a = -2f$. Therefore, $g = f = -a/2$.
Final General Equation
Substituting $g = f = -a/2$ into the general equation: $x^2 + y^2 + 2(-a/2)x + 2(-a/2)y = 0$ $x^2 + y^2 - ax - ay = 0$
Where 'a' is a constant representing the intercept length on both axes.
Summary
- If the circle passes through the origin, $c=0$.
- For equal intercepts $a$ on both axes, the equation simplifies to the form $x^2 + y^2 - ax - ay = 0$.
- This represents a family of circles where the center is at $(a/2, a/2)$ and the radius is $\sqrt{(a/2)^2 + (a/2)^2} = \frac{|a|}{\sqrt{2}}$.