Conic Sections (Parabola, Ellipse & Hyperbola) All Important Formulas & Tricks for JEE Mains 2023

Summary

This educational video provides a comprehensive revision of key concepts and formulas in conic sections, focusing on essential results needed for competitive exams like JEE. The content covers parabolas, ellipses, and hyperbolas, explaining their equations, properties, and the relationship between their parameters. It also highlights the important concepts of tangents, normals, and focal chords, detailing various forms of their equations. Through clear explanations and examples, the video aims to simplify complex topics and help students grasp essential geometric relationships for solving problems efficiently.

Takeaways

• 😀 Parabolas are one of the most frequently tested topics in the GE exam, with key formulas like the standard rightward opening parabola (y^2 = 4ax) and leftward opening parabola (y^2 = -4x).
• 😀 The focal properties of a parabola include the coordinates of the focus (a, 0), directrix equation (x = ±a), and the length of the lattice rectum (4a).
• 😀 The parametric coordinates of any point on a parabola are given by (at^2, 2at) for a rightward-opening parabola.
• 😀 The concept of focal chords is crucial, where the product of the parameters of the ends of a focal chord is always equal to 1.
• 😀 The equation of the tangent to a parabola can be written in point form, parametric form, and slope form, each with a distinct method of calculation.
• 😀 The equation of normal to a parabola follows similar forms as the tangent equation, and its slope is the negative reciprocal of the tangent's slope.
• 😀 An ellipse with its center at the origin has the equation x^2/a^2 + y^2/b^2 = 1, with foci at ±ae, and a directrix at x = ±a/e.
• 😀 The parametric coordinates of any point on an ellipse are (a cos θ, b sin θ), and the relationship between a, b, and eccentricity e is crucial for solving problems.
• 😀 For ellipses, the equation of tangent can be expressed in point form (T = 0), parametric form (substituting cos θ and sin θ), and slope form (C = ±√(a²m² + b²)).
• 😀 A hyperbola's equation can be in the form x²/a² - y²/b² = 1, with important relationships between a, b, and eccentricity e, where e > 1.
• 😀 The equation of the tangent to a hyperbola also has point form, parametric form, and slope form, with similar principles as for ellipses, but with adjustments for the hyperbolic curve.

Q & A

What is the standard equation of a rightward opening parabola?

The standard equation of a rightward opening parabola with its vertex at the origin is given by y² = 4ax, where 'a' represents the distance from the vertex to the focus, as well as the distance from the vertex to the directrix.

What does the parameter 'a' represent in the equation of a rightward opening parabola?

In the equation y² = 4ax of a rightward opening parabola, 'a' represents the distance between the vertex and the focus, as well as the distance from the vertex to the directrix.

What is the relationship between the parameters of the extremities of a focal chord in a parabola?

For a focal chord PQ of a parabola, if the parameters of points P and Q are T and T', respectively, then the product of the parameters of the extremities of the focal chord is always equal to 1, i.e., T × T' = 1.

What is the equation of the tangent to a parabola in point form?

The equation of the tangent to a parabola y² = 4ax in point form, where (x₁, y₁) is the point of tangency, is given by yy₁ = 2a(x + x₁). This is derived by substituting the coordinates of the point of tangency into the equation.

What is the significance of the quantity '4a' in the equation of a parabola?

In the equation of a parabola y² = 4ax, the quantity '4a' represents the length of the lattice rectum, which is the distance between the points where the parabola intersects the line perpendicular to the axis of symmetry through the focus.

How does the equation of a parabola change when it opens leftward instead of rightward?

The equation of a parabola that opens leftward, with the vertex at the origin, is y² = -4x. This is simply the reflection of the rightward opening parabola y² = 4ax about the y-axis, and it maintains the same properties.

What are the parametric coordinates of a point on a rightward opening parabola?

The parametric coordinates of a point on the rightward opening parabola y² = 4ax are given by (at², 2at), where 't' is the parameter representing a real value.

What is the equation of the normal to a parabola in slope form?

The equation of the normal to a parabola y² = 4ax in slope form is given by y = mx + C, where the y-intercept C is equal to -2am - am³, with 'm' being the slope of the tangent.

What is the standard equation of a horizontal ellipse?

The standard equation of a horizontal ellipse with its center at the origin is x²/a² + y²/b² = 1, where 'a' and 'b' represent the lengths of the semi-major and semi-minor axes, respectively.

How is the relationship between a, b, and the eccentricity e of an ellipse expressed?

For an ellipse, the relationship between the semi-major axis 'a', semi-minor axis 'b', and the eccentricity 'e' is given by the equation b² = a²(1 - e²), where e is the eccentricity of the ellipse.