Exercise Questions

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

Key Concepts

This exercise focuses on the following concepts:

• Intersection of lines and circles

• Tangents to circles (equations and conditions)

• Normals to circles

• Condition for tangency: distance from center equals radius
• Perpendicularity of tangent and radius
• Chords and diameters of circles
• Condition for two circles to touch (internally or externally)

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Important Formulas

Below are the key formulas used in this exercise:

Standard Circle Equation: $(x-h{)}^2+(y-k{)}^2=r^2$

General Circle Equation: $x^2+y^2+2gx+2fy+c=0$ with center $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$

Condition for Line to be Tangent to Circle: $d=r$ where $d$ is perpendicular distance from center to line

Equation of Tangent to Circle $x^2+y^2=a^2$ at $(x_1,y_1)$: $x{x}_1+y{y}_1=a^2$

Equation of Normal to Circle at $(x_1,y_1)$: $y-y_1=\frac{y_1}{x_1}(x-x_1)$

Condition for Two Circles to Touch: $|C_1{C}_2|=r_1\pm r_2$ (external: $+$ , internal: $-$)

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Summary

This exercise covers fundamental properties of circles and their interactions with lines. Key strategies include: converting general form to standard form to identify center and radius; using the distance formula to establish tangency conditions; applying the geometric property that tangent is perpendicular to radius; and using the fact that the normal passes through the center. For intersection problems, simultaneous equations are essential.

When circles touch, comparing the distance between centers with sum/difference of radii determines the nature of contact.