Find the equation of the circle passing through $(0, 0)$ and making intercepts $a$ and $b$ on the coordinate axes.

Question

Accepted Answer

> [!TLDR]
> The circle passes through the origin and cuts the x‑axis at length $a$ and the y‑axis at length $b$. Using the intercept form of a circle, the required equation is $x^2 + y^2 - ax - by = 0$.

## Why this approach works

When a circle passes through the origin and makes intercepts on the axes, those intercepts are not just lengths — they tell us exactly where the circle meets the axes. If the circle cuts the x‑axis at a distance $a$ from the origin, the two intersection points are $(0,0)$ and $(a,0)$. Similarly, on the y‑axis the points are $(0,0)$ and $(0,b)$.

So we actually know three points on the circle: $(0,0)$, $(a,0)$, and $(0,b)$. Three points uniquely determine a circle. The neatest way to handle this is to use the **general equation of a circle** and plug these points in.

> [!FORMULA]
> The general equation of a circle is:
> $$x^2 + y^2 + 2gx + 2fy + c = 0$$
> where centre is $(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$.

Let’s work through it step by step.

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1. **Use the fact that $(0,0)$ lies on the circle**  
   Substitute $x = 0$, $y = 0$ into $x^2 + y^2 + 2gx + 2fy + c = 0$:  
   $$0 + 0 + 0 + 0 + c = 0 \quad\Rightarrow\quad c = 0$$  
   So the equation simplifies to $x^2 + y^2 + 2gx + 2fy = 0$.

2. **Use the x‑intercept condition**  
   The circle meets the x‑axis at $(a,0)$. Substitute $x = a$, $y = 0$:  
   $$a^2 + 0 + 2g a + 0 = 0 \quad\Rightarrow\quad a^2 + 2g a = 0$$  
   Since $a 
eq 0$ (otherwise the intercept would be zero, a degenerate case), divide by $a$:  
   $$a + 2g = 0 \quad\Rightarrow\quad g = -\frac{a}{2}$$

3. **Use the y‑intercept condition**  
   The circle meets the y‑axis at $(0,b)$. Substitute $x = 0$, $y = b$:  
   $$0 + b^2 + 0 + 2f b = 0 \quad\Rightarrow\quad b^2 + 2f b = 0$$  
   With $b 
eq 0$, divide by $b$:  
   $$b + 2f = 0 \quad\Rightarrow\quad f = -\frac{b}{2}$$

4. **Write the final equation** …

Q.Find the equation of the circle passing through $(0, 0)$ and making intercepts $a$ and $b$ on the coordinate axes.

Concept understanding — Circle Equation Standard Form

Where the Circle Equation Comes From

From Geometry to Algebra

What Each Piece Tells You

Quick Example

Why this approach works

Q.Find the equation of the circle passing through (0,0)(0, 0)(0,0) and making intercepts aaa and bbb on the coordinate axes.

Why this approach works

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Q.Find the equation of the circle passing through $(0, 0)$ and making intercepts $a$ and $b$ on the coordinate axes.

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