Mathematics: Circles Notes for JEE

Standard Equation of a Circle

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

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): $${\bf (y - k) = \frac{y_1 - k}{x_1 - h} (x - h)}$$
• 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: $${\bf (x_1 - h)^2 + (y_1 - k)^2 - r^2}$$
• 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: $${\bf x^2 + y^2 + 2gx + 2fy + c = 0}$$
• Finding the center and radius of a circle from its equation

Area and Circumference of Circles

• Area: $${\bf A = \pi r^2\hspace{0.5}cm}$$
• Circumference: $${\bf C = 2\pi r\hspace{0.5}cm}$$

Parametric Equations of Circles

• Parametric equations of a circle with center (h, k) and radius r: $${\bf x = h + r\cos \theta, \hspace{0.5}cm y = k + r\sin \theta,\hspace{0.5}cm 0 \leq \theta \leq 2\pi}$$

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.