Equations of Tangent, Normal and Chord of a Circle (JEE Advanced)

“equations of tangent, normal and chord”

Source: JEE Advanced 2026 Information Bulletin, Annexure-I. Closest JEE Main topic: A standard form of equations of a circle, the general form of the equation of a circle.

JEE Main focuses on basic circle equations, but Advanced expects you to manipulate tangent and normal equations, find chord lengths, and solve problems involving contact conditions. These concepts appear in combined problems with conics and locus, which Main rarely tests.

• A tangent to a circle is perpendicular to the radius at the point of contact, which gives you the slope of the tangent directly
• The equation of tangent at point (x₁, y₁) on circle x² + y² = a² is xx₁ + yy₁ = a², obtained by replacing x² with xx₁ and y² with yy₁
• For a line y = mx + c to be tangent to x² + y² = a², the condition is c² = a²(1 + m²), meaning c = ±a√(1 + m²)
• The chord of contact from an external point (x₁, y₁) to circle x² + y² = a² has equation xx₁ + yy₁ = a², same form as tangent equation
• Length of chord cut by a line from a circle can be found using: 2√(r² − d²), where d is perpendicular distance from center to the line
• Two tangents from an external point are equal in length, and this length equals √(S₁) where S₁ is the value of circle equation at that point

• Tangent at (x₁, y₁) on x² + y² = a²: xx₁ + yy₁ = a²
• Tangent at (x₁, y₁) on x² + y² + 2gx + 2fy + c = 0: xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
• Tangent with slope m to x² + y² = a²: y = mx ± a√(1 + m²)
• Normal at (x₁, y₁) on x² + y² = a²: xy₁ − yx₁ = 0 (passes through origin)
• Chord of contact from (x₁, y₁): xx₁ + yy₁ = a² (for circle x² + y² = a²)
• Length of tangent from (x₁, y₁) to circle: √(x₁² + y₁² − a²)
• Chord length = 2√(r² − d²), where d = perpendicular distance from center to chord
• Equation of chord with midpoint (h, k): T = S₁, i.e., xh + yk = h² + k² (for x² + y² = a²)

• Confusing tangent at a point ON the circle with tangent FROM an external point. The formulas look similar but the situations are different.
• Forgetting the ± sign in y = mx ± a√(1 + m²). There are always two tangents with a given slope, one on each side of the circle.
• Using the tangent formula for x² + y² = a² directly on a general circle without shifting to center form or using the general formula.
• Calculating chord length as √(r² − d²) instead of 2√(r² − d²). The formula gives half the chord, so you must double it.

Question. Find the equations of tangents to the circle x² + y² = 25 which are parallel to the line 3x − 4y + 7 = 0.

Solved twice independently; both solutions agree.