Exercise Questions

All questions in this exercise are listed below. Click on a question to view its solution.

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Key Concepts

This exercise focuses on the following concepts:

Standard and General Equations of a Circle

A circle is the locus of all points equidistant from a fixed point called the centre.

Standard Form: $(x-h{)}^2+(y-k{)}^2=r^2$ where $(h,k)$ is the centre and $r>0$ is the radius.

General Form: $x^2+y^2+2gx+2fy+c=0$

• Centre: $(-g,-f)$
• Radius: $r=\sqrt{g^2+f^2-c}$ (real circle requires $g^2+f^2-c>0$)

Convert between forms by completing the square: $(x+g{)}^2+(y+f{)}^2=g^2+f^2-c$

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Centre and Radius Determination

• From standard form: read off $(h,k)$ and $r$ directly.
• From two diameters: the centre is the intersection of the two diameter lines — solve the linear system simultaneously.
• From general form: centre $=(-g,-f)$, radius $=\sqrt{g^2+f^2-c}$.

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Circle Passing Through Points

Three non-collinear points: The centre is equidistant from all three points. Set $(h-x_1{)}^2+(k-y_1{)}^2=(h-x_2{)}^2+(k-y_2{)}^2=(h-x_3{)}^2+(k-y_3{)}^2$ and solve the resulting linear system for $(h,k)$, then find $r$.

Two points with centre on a given line $L$: Equate the distances from the centre to both points (both equal $r$), and substitute the centre into the equation of $L$. Solve the system.

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Tangency Conditions

A line $ax+by+c=0$ is tangent to a circle with centre $(h,k)$ and radius $r$ if and only if: $d=\frac{|ah+bk+c|}{\sqrt{a^2+b^2}}=r$

Additionally, the radius drawn to the point of tangency is perpendicular to the tangent line (product of slopes $=-1$). These two conditions together determine unknown parameters.

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Circles Touching Coordinate Axes

If a circle of radius $r$ touches both coordinate axes, then $|h|=|k|=r$:

• First quadrant: centre $(r,r)$
• Second quadrant: centre $(-r,r)$
• Third quadrant: centre $(-r,-r)$
• Fourth quadrant: centre $(r,-r)$

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Diameters of a Circle

A diameter is a chord passing through the centre. Given two diameter equations, the centre is their point of intersection.

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Family of Circles Through Intersection of Two Circles

If $S_1=0$ and $S_2=0$ are two circles, then $S_1+\lambda S_2=0(\lambda \ne -1)$ represents a family of circles passing through their points of intersection.

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Important Formulas Summary

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Summary

This exercise covers methods for determining the equation of a circle under various geometric constraints. Key strategies include: using standard form when centre and radius are known; converting between general and standard forms by completing the square; applying the tangency condition ($d=r$) for problems involving tangent lines; solving systems of equations to find unknown parameters using given points, tangent conditions, or diameter equations. Special cases include circles touching both axes ($|h|=|k|=r$) and the family of circles through the intersection of two given circles.