MathsMediumClass 12

Chord of Contact

Conic Sections

JEE Qs

Hard

min

Master the consistent application of the T=0 form for the chord of contact across all conics, as it is a frequently tested concept and a gateway to advanced topics like pole and polar.

🧮 Key Formulas

General form for a conic S=0 (e.g., ax² + by² + 2hxy + 2gx + 2fy + c = 0): T=0, where T is obtained by replacing x² by x x₁, y² by y y₁, 2xy by x y₁ + y x₁, 2x by x + x₁, 2y by y + y₁ in S=0.

For Circle x² + y² = a²: x x₁ + y y₁ = a²

For Parabola y² = 4ax: y y₁ = 2a(x + x₁)

For Ellipse x²/a² + y²/b² = 1: x x₁ / a² + y y₁ / b² = 1

For Hyperbola x²/a² - y²/b² = 1: x x₁ / a² - y y₁ / b² = 1

✅ Key Points for JEE

1The equation of the chord of contact of tangents drawn from an external point (x₁, y₁) to any conic S=0 is given by T=0.
2Geometrically, the chord of contact is the straight line segment joining the two points where the tangents from the external point (x₁, y₁) touch the conic.
3The 'T=0' form is extremely versatile and applies universally to all standard conics (circle, parabola, ellipse, hyperbola), making it a unified concept.
4Understanding the chord of contact is fundamental for grasping the concept of 'Pole and Polar', as the chord of contact is essentially the polar of the external point.

⚠️ Common Mistakes

✕Confusing the chord of contact equation (T=0 for an external point) with the tangent equation at a point (T=0 for a point on the conic).
✕Incorrectly identifying the external point (x₁, y₁) in problems, leading to errors in applying the T=0 formula.
✕Algebraic errors during substitution of coordinates or simplification of the resulting linear equation.
✕Not recognizing that the chord of contact is a straight line, and problems might involve its length, slope, or intersection properties.

📝 Practice Questions

See all

Q70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1

2019·MCQHard

Q69.If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P , then the distance of P from the origin (units), is: + (1) 2(3 2√2) (2) 3 + 2√2 + (3) √2 + 1 (4) 2(√2 1) JEE Main 2017 (08 Apr Online) JEE Main Previous Year Paper

2017·Multi conceptHard

NCERT Chapters

Class 11 Maths Ch 11: Conic Sections