MathsMediumClass 12

Tangent & Normal to Ellipse

Conic Sections

JEE Qs

Hard

min

Thoroughly understand the derivations of all tangent and normal forms, especially the slope form, as it is crucial for solving problems involving common tangents, locus, and director circle.

🧮 Key Formulas

Standard equation of ellipse: x^2/a^2 + y^2/b^2 = 1

Slope of tangent at (x1, y1) on ellipse: dy/dx = -b^2x1 / (a^2y1)

Equation of tangent at point P(x1, y1): x*x1/a^2 + y*y1/b^2 = 1

Equation of tangent in parametric form at P(a cosθ, b sinθ): x/a cosθ + y/b sinθ = 1

Equation of tangent in slope form (y = mx + c): y = mx ± sqrt(a^2m^2 + b^2)

Condition for y = mx + c to be tangent to ellipse: c^2 = a^2m^2 + b^2

Equation of normal at point P(x1, y1): a^2x/x1 - b^2y/y1 = a^2 - b^2

Equation of normal in parametric form at P(a cosθ, b sinθ): ax secθ - by cosecθ = a^2 - b^2

Director Circle (locus of intersection of perpendicular tangents): x^2 + y^2 = a^2 + b^2

✅ Key Points for JEE

1Master the three forms (point, parametric, slope) of tangent and normal equations and identify which form is most efficient for a given problem.
2The condition of tangency (c^2 = a^2m^2 + b^2) is fundamental for problems involving common tangents, tangents with a given slope, or locus of points.
3Understand the director circle property: the locus of the point of intersection of two perpendicular tangents to an ellipse is a circle x^2 + y^2 = a^2 + b^2.
4Recall the optical property: the tangent at any point on an ellipse makes equal angles with the focal radii to that point (i.e., it is the external angle bisector of the angle formed by the focal radii).
5Normal is perpendicular to the tangent at the point of contact; its slope is -1/(slope of tangent).

⚠️ Common Mistakes

✕Sign errors, especially in distinguishing tangent/normal equations for ellipse vs. hyperbola, or missing the '±' in slope form of tangent.
✕Algebraic errors when manipulating equations to find slopes, intercepts, or specific points.
✕Incorrectly applying the condition for tangency or forgetting to check for special cases (e.g., vertical tangents/normals).
✕Confusing the role of 'a' and 'b' when the major axis is along the y-axis (i.e., x^2/b^2 + y^2/a^2 = 1).

📝 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
Class 12 Maths Ch 6: Application of Derivatives