Parametric Form — (at², 2at)
Conic Sections
8
JEE Qs
8%
Hard
60
min
Master the parametric equations for tangents, normals, and chords, along with their key properties (like t1t2 = -1), as this significantly speeds up problem-solving compared to Cartesian methods.
🧮 Key Formulas
✅ Key Points for JEE
- 1The parameter 't' provides a powerful way to represent any point on the parabola y^2 = 4ax, simplifying derivations and calculations for geometric properties.
- 2The slope of the tangent at the point (at^2, 2at) is 1/t. This is a crucial shortcut for problems involving tangents.
- 3The intersection point of tangents drawn at points 't1' and 't2' is (at1t2, a(t1 + t2)). If these tangents are perpendicular, then t1t2 = -1, and their intersection lies on the directrix.
- 4If a chord joining points 't1' and 't2' passes through the focus (a, 0), then the condition is t1t2 = -1. This defines a focal chord.
- 5For three normals drawn from a point (h, k) to the parabola y^2 = 4ax, the parameters t1, t2, t3 satisfy t1 + t2 + t3 = 0, which is useful for problems involving co-normal points.
⚠️ Common Mistakes
- ✕Confusing the specific parametric form (at^2, 2at) for y^2 = 4ax with other forms for parabolas (e.g., x^2 = 4ay) or other conic sections.
- ✕Incorrectly deriving or recalling the equations for tangent, normal, or chord in parametric form, especially mixing up coefficients or powers of 't' and 'a'.
- ✕Failing to recognize the significance of conditions like t1t2 = -1 (for perpendicular tangents or focal chords) or applying them incorrectly.
📝 Practice Questions
See allQ70.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
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
NCERT Chapters
- Class 11 Mathematics Ch 11: Conic Sections