The equation of the circle passing through (0, 0) and which makes intercepts a and b on the coordinates axes is?

Question image

Understand the Problem

The question is asking us to find the equation of a circle that passes through the origin (0,0) and has specific intercepts 'a' and 'b' on the coordinate axes. It provides multiple choice options for the possible equations.

Answer

The equation is given by $x^2 + y^2 - ax - by = 0$.
Answer for screen readers

The equation of the circle is given by:

$$ x^2 + y^2 - ax - by = 0 $$

Steps to Solve

  1. General Form of Circle Equation

The general equation of a circle can be expressed as:

$$ x^2 + y^2 + Dx + Ey + F = 0 $$

Since the circle passes through the origin (0, 0), we can substitute these coordinates into the equation:

$$ 0^2 + 0^2 + D \cdot 0 + E \cdot 0 + F = 0 $$

This simplifies to:

$$ F = 0 $$

Thus, the equation simplifies to:

$$ x^2 + y^2 + Dx + Ey = 0 $$

  1. Understanding Circle Intercepts

A circle intercepts the x-axis at points (a, 0) and (-a, 0), and intercepts the y-axis at points (0, b) and (0, -b).

  1. Substituting Intercepts into the Equation

From the intercepts, we can determine that:

  • When $y = 0$: The equation becomes $x^2 + Dx = 0$, which factors to $x(x + D) = 0$, giving intercepts at $x = 0$ and $x = -D$.

  • Therefore, the relationship is that $a = -D$ (x-intercept at $(a, 0)$).

  • When $x = 0$: The equation becomes $y^2 + Ey = 0$, which also factors to $y(y + E) = 0$, giving intercepts at $y = 0$ and $y = -E$.

  • Thus, the relationship is that $b = -E$ (y-intercept at $(0, b)$).

  1. Final Formulation

Replacing $D$ and $E$ with $-a$ and $-b$ respectively, we get:

$$ x^2 + y^2 - ax - by = 0 $$

This matches option (d) from the provided choices.

The equation of the circle is given by:

$$ x^2 + y^2 - ax - by = 0 $$

More Information

The equation $x^2 + y^2 - ax - by = 0$ represents a circle that passes through the origin (0,0) and has intercepts of length 'a' and 'b' on the x-axis and y-axis respectively. This type of problem is common in coordinate geometry and highlights the relationship between intercepts and the coefficients in the circle's equation.

Tips

  • Confusing the signs of coefficients: Always remember that the intercept corresponds to negative values of the coefficients when going from intercept form to general circle equations.
  • Forgetting to substitute correctly when plugging in the origin (0,0). Always ensure that the equation balances when plugging in specific points.

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