Question:
The equation of a circle that passes through the origin and cut off intercepts $-2$ and $3$ on the X-axis and Y-axis respectively is
The equation of a circle that passes through the origin and cut off intercepts $-2$ and $3$ on the X-axis and Y-axis respectively is
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For any circle passing through the origin and cutting intercepts $a$ and $b$ on the axes, its equation can be directly written as $x^2 + y^2 - ax - by = 0$. Here, $a = -2$ and $b = 3$, giving $x^2 + y^2 - (-2)x - (3)y = 0 \Rightarrow x^2 + y^2 + 2x - 3y = 0$ instantly!
Updated On: Jun 18, 2026
- $x^2 + y^2 - 2x + 3y = 0$
- $x^2 + y^2 + 2x + 3y = 0$
- $x^2 + y^2 + 2x - 3y = 0$
- $x^2 + y^2 - 2x - 3y = 0$
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We need to find the equation of a circle passing through three known points on the Cartesian plane: the origin $(0,0)$, the X-intercept point $(-2,0)$, and the Y-intercept point $(0,3)$.
Step 2: Key Formula or Approach:
The general equation of a circle is given by: $$x^2 + y^2 + 2gx + 2fy + c = 0$$ We will substitute the three coordinates into this general equation to find the values of the constants $g, f,$ and $c$.
Step 3: Detailed Explanation:
1. Since the circle passes through the origin $(0,0)$: $$0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \Rightarrow c = 0$$ The equation simplifies to $x^2 + y^2 + 2gx + 2fy = 0$.
2. The circle passes through $(-2,0)$ on the X-axis: $$(-2)^2 + 0^2 + 2g(-2) + 2f(0) = 0$$ $$4 - 4g = 0 \Rightarrow g = 1$$ 3. The circle passes through $(0,3)$ on the Y-axis: $$0^2 + 3^2 + 2g(0) + 2f(3) = 0$$ $$9 + 6f = 0 \Rightarrow f = -\frac{3}{2}$$ Substituting $g = 1$, $f = -\frac{3}{2}$, and $c = 0$ into the general equation: $$x^2 + y^2 + 2(1)x + 2\left(-\frac{3}{2}\right)y = 0$$ $$x^2 + y^2 + 2x - 3y = 0$$
Step 4: Final Answer:
The correct circle equation is $x^2 + y^2 + 2x - 3y = 0$, which perfectly matches option (C).
We need to find the equation of a circle passing through three known points on the Cartesian plane: the origin $(0,0)$, the X-intercept point $(-2,0)$, and the Y-intercept point $(0,3)$.
Step 2: Key Formula or Approach:
The general equation of a circle is given by: $$x^2 + y^2 + 2gx + 2fy + c = 0$$ We will substitute the three coordinates into this general equation to find the values of the constants $g, f,$ and $c$.
Step 3: Detailed Explanation:
1. Since the circle passes through the origin $(0,0)$: $$0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \Rightarrow c = 0$$ The equation simplifies to $x^2 + y^2 + 2gx + 2fy = 0$.
2. The circle passes through $(-2,0)$ on the X-axis: $$(-2)^2 + 0^2 + 2g(-2) + 2f(0) = 0$$ $$4 - 4g = 0 \Rightarrow g = 1$$ 3. The circle passes through $(0,3)$ on the Y-axis: $$0^2 + 3^2 + 2g(0) + 2f(3) = 0$$ $$9 + 6f = 0 \Rightarrow f = -\frac{3}{2}$$ Substituting $g = 1$, $f = -\frac{3}{2}$, and $c = 0$ into the general equation: $$x^2 + y^2 + 2(1)x + 2\left(-\frac{3}{2}\right)y = 0$$ $$x^2 + y^2 + 2x - 3y = 0$$
Step 4: Final Answer:
The correct circle equation is $x^2 + y^2 + 2x - 3y = 0$, which perfectly matches option (C).
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