Parabola — Tangent Normal Chord
Parabola
42
JEE Qs
30%
Hard
80
min
Prioritize understanding the derivation and applications of parametric forms for tangents and normals, as they are key to solving complex problems efficiently.
🧮 Key Formulas
✅ Key Points for JEE
- 1Mastering the parametric form (at^2, 2at) for parabola is crucial as it often simplifies tangent and normal problems significantly by reducing variables.
- 2Understand the distinction and correct application of T=S1 for a chord with a given midpoint versus the chord of contact from an external point.
- 3Remember key geometrical properties: the foot of the perpendicular from the focus to any tangent lies on the directrix, and tangents drawn at the extremities of a focal chord intersect on the directrix at right angles.
- 4The locus of the point of intersection of perpendicular tangents to a parabola is its directrix (this is the parabola's Director Circle).
- 5For problems involving multiple tangents or normals, the parametric equations for the points of contact and their intersection points are invaluable.
⚠️ Common Mistakes
- ✕Confusing the formulas for tangent and normal for different conic sections or using the wrong form (point, slope, parametric) in a given situation.
- ✕Incorrectly applying the T=S1 relation, leading to errors when distinguishing between a chord of contact and a chord with a given midpoint.
- ✕Algebraic errors, especially with signs and coefficients 'a' and 't', which are common when solving complex problems involving multiple equations.
- ✕Not checking the condition m != 0 when using the slope form (y = mx + a/m), which is invalid for vertical tangents.
📝 Practice Questions
See allQ24.The focus of the parabola y2 = 4x + 16 is the centre of the circle C of radius 5 . If the values of λ, for which C passes through the point of intersection of the lines 3x −y = 0 and x + λy = 4, are λ1 and λ2, λ1 < λ2 , then 12λ1 + 29λ2 is equal to
Q3. Let ABCD be a trapezium whose vertices lie on the parabola y2 = 4x. Let the sides AD and BC of the trapezium be parallel to y -axis. If the diagonal AC is of length 25 and it passes through the point (1, 0), then 4 the area of ABCD is (1) 75 4 (2) = 252 (3) 125 (4) 75 8 8
Q9. Let P(4, 4√3) be a point on the parabola y2 = 4ax and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to : 2025 (22 Jan Shift 2) JEE Main Previous Year Paper (1) 17√3 (2) 263√3 8 (3) 34√3 (4) 343√3 3 8 π
Q19.If in the expansion of (1 + x)p(1 −x)q , the coefficients of x and x2 are 1 and -2 , respectively, then p2 + q2 is equal to : (1) 18 (2) 13 (3) 8 (4) 20 a
Q19.If the equation of the parabola with vertex V ( 32 , 3) and the directrix x + 2y = 0 is αx2 + βy2 −γxy −30x −60y + 225 = 0, then α + β + γ is equal to : ∣ ∣ 2025 (24 Jan Shift 2) JEE Main Previous Year Paper (1) 7 (2) 9 (3) 8 (4) 6 (1+β2) (1+γ2) (1+α2) is + + +
Q21.Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B , and a denotes the area of △SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is -
NCERT Chapters
- Class 11 Maths Ch 11: Conic Sections