Mathematics: Circles Notes for JEE

Standard Equation of a Circle

• Equation of a circle with center (h, k) and radius r: $$\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{h}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{(}\mathbf{y}\mathbf{-}\mathbf{k}{\mathbf{)}}^{\mathbf{2}}\mathbf{=}{\mathbf{r}}^{\mathbf{2}}$$

Properties of Chords

• Chords intersecting at right angles:
• Product of lengths of segments of one chord from the point of intersection is equal to the product of lengths of segments of the other chord.
• Tangents from an external point:
• Tangents drawn from an external point to a circle are equal in length.
• Lengths and positions of chords:
• The perpendicular from the center of the circle to a chord bisects the chord.

Tangents

• Equation of tangent at point (x₁, y₁) on circle with center (h, k): $$\mathbf{(}\mathbf{y}\mathbf{-}\mathbf{k}\mathbf{)}\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{1}}\mathbf{-}\mathbf{k}}{{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{h}}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{h}\mathbf{)}$$
• Conditions for tangency:
• Line has the same slope as the radius at the point of tangency.
• Line passes through the center of the circle.

Angle Properties

• Angle formed by two tangents from an external point: Equal
• Inscribed angles:
  • Measure of an inscribed angle is half of the measure of the intercepted arc.
  • Opposite angles in a cyclic quadrilateral are supplementary.

Cyclic Quadrilaterals

• Properties of cyclic quadrilaterals:
  • Opposite angles are supplementary.
  • Opposite sides multiply to give the same product.
• Condition for a quadrilateral to be cyclic:
• Sums of opposite angles are equal.

Power of a Point

• Power of a point (x₁, y₁) with respect to a circle with center (h, k) and radius r: $$\mathbf{(}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{h}{\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{(}{\mathbf{y}}_{\mathbf{1}}\mathbf{-}\mathbf{k}{\mathbf{)}}^{\mathbf{2}}\mathbf{-}{\mathbf{r}}^{\mathbf{2}}$$
• If the power of a point is positive, the point is outside the circle.
• If the power of a point is zero, the point lies on the circle.
• If the power of a point is negative, the point is inside the circle.

Equations of Circles

• General equation of a circle: $${\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{g}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{f}\mathbf{y}\mathbf{+}\mathbf{c}\mathbf{=}\mathbf{0}$$
• Finding the center and radius of a circle from its equation

Area and Circumference of Circles

• Area: $$\text{Missing dimension or its units for hspace}$$
• Circumference: $$\text{Missing dimension or its units for hspace}$$

Parametric Equations of Circles

• Parametric equations of a circle with center (h, k) and radius r: $$\text{Missing dimension or its units for hspace}$$

Intersection of Circles and Lines

• Points of intersection:
• Solve the equations of the circle and line simultaneously to find the points of intersection.
• Tangency conditions:
• The distance from the center of the circle to the line is equal to the radius.
• The line is perpendicular to the radius at the point of tangency.

Recommended Resources

• NCERT Mathematics textbooks for Class 11 and Class 12.
• JEE Main and Advanced Previous Year Question Papers and Sample Papers.
• Reference books and study guides specifically for the JEE exam.
• Online resources and video tutorials related to Circle topics.