MathsMediumClass 12

Tangent & Normal — Conditions, equations

Conic Sections

JEE Qs

Hard

min

Master all three forms (point, slope, parametric) of tangent and normal equations for each conic, as the most efficient approach often depends on the specific problem's context.

🧮 Key Formulas

Equation of Tangent to y=f(x) at (x1, y1): y - y1 = (dy/dx)|_(x1,y1) * (x - x1)

Equation of Normal to y=f(x) at (x1, y1): y - y1 = -1 / (dy/dx)|_(x1,y1) * (x - x1)

Condition for y=mx+c to be tangent to Circle x^2 + y^2 = a^2: c = ±a*sqrt(1+m^2)

Tangent to Circle x^2 + y^2 = a^2 at (x1, y1): x*x1 + y*y1 = a^2

Tangent to Circle x^2 + y^2 = a^2 in Slope form: y = mx ± a*sqrt(1+m^2)

Normal to Circle x^2 + y^2 = a^2 at (x1, y1): y*x1 - x*y1 = 0 (passes through origin)

Condition for y=mx+c to be tangent to Parabola y^2 = 4ax: c = a/m

Tangent to Parabola y^2 = 4ax at (x1, y1): y*y1 = 2a(x + x1)

Tangent to Parabola y^2 = 4ax in Slope form: y = mx + a/m

Tangent to Parabola y^2 = 4ax in Parametric form (at^2, 2at): yt = x + at^2

Normal to Parabola y^2 = 4ax at (x1, y1): y - y1 = (-y1 / 2a)(x - x1)

Normal to Parabola y^2 = 4ax in Parametric form (at^2, 2at): y + tx = 2at + at^3

Condition for y=mx+c to be tangent to Ellipse x^2/a^2 + y^2/b^2 = 1: c^2 = a^2*m^2 + b^2

Tangent to Ellipse x^2/a^2 + y^2/b^2 = 1 at (x1, y1): x*x1/a^2 + y*y1/b^2 = 1

Tangent to Ellipse x^2/a^2 + y^2/b^2 = 1 in Slope form: y = mx ± sqrt(a^2*m^2 + b^2)

Tangent to Ellipse x^2/a^2 + y^2/b^2 = 1 in Parametric form (a*cosθ, b*sinθ): x*cosθ/a + y*sinθ/b = 1

Normal to Ellipse x^2/a^2 + y^2/b^2 = 1 at (x1, y1): a^2*x/x1 - b^2*y/y1 = a^2 - b^2

Normal to Ellipse x^2/a^2 + y^2/b^2 = 1 in Parametric form (a*cosθ, b*sinθ): a*x*secθ - b*y*cosecθ = a^2 - b^2

Condition for y=mx+c to be tangent to Hyperbola x^2/a^2 - y^2/b^2 = 1: c^2 = a^2*m^2 - b^2

Tangent to Hyperbola x^2/a^2 - y^2/b^2 = 1 at (x1, y1): x*x1/a^2 - y*y1/b^2 = 1

Tangent to Hyperbola x^2/a^2 - y^2/b^2 = 1 in Slope form: y = mx ± sqrt(a^2*m^2 - b^2)

Tangent to Hyperbola x^2/a^2 - y^2/b^2 = 1 in Parametric form (a*secθ, b*tanθ): x*secθ/a - y*tanθ/b = 1

Normal to Hyperbola x^2/a^2 - y^2/b^2 = 1 at (x1, y1): a^2*x/x1 + b^2*y/y1 = a^2 + b^2

Normal to Hyperbola x^2/a^2 - y^2/b^2 = 1 in Parametric form (a*secθ, b*tanθ): a*x*cosθ + b*y*cotθ = a^2 + b^2

Equation of Chord of Contact (T=0 form for an external point (x1, y1)): S1 = x*x1 + y*y1 - a^2 = 0 for circle x^2+y^2=a^2, similar for other conics.

✅ Key Points for JEE

1The derivative dy/dx gives the slope of the tangent at a given point on a curve; the slope of the normal is the negative reciprocal.
2The 'T=0' method (replacing x^2 with x*x1, y^2 with y*y1, 2x with x+x1, 2y with y+y1, 2xy with x*y1+y*x1) directly gives the equation of the tangent at a point (x1, y1) on the conic.
3For problems involving tangents from an external point, either use the slope form (y=mx+c) with the condition of tangency or assume the point of tangency (x1, y1) and use the T=0 form.
4Normal to a circle always passes through its center.
5Understanding the specific conditions of tangency (c = a/m for parabola, c^2 = a^2m^2+b^2 for ellipse, etc.) is crucial for solving problems involving tangent lines of a given slope.

⚠️ Common Mistakes

✕Confusing the signs in slope forms or normal equations, especially for hyperbola and ellipse.
✕Incorrectly calculating dy/dx for implicit functions or making algebraic errors in simplification.
✕Applying the T=0 method for a point not on the curve (T=0 is for tangent at a point on the curve, or chord of contact from an external point, or polar of a point).
✕Forgetting that for a normal, the product of slopes of tangent and normal is -1, especially when dealing with vertical tangents (horizontal normals) or horizontal tangents (vertical normals).

📝 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

🚀 This Unlocks

Locus Problems