JEE Hyperbola Previous Year Questions With Solutions

JEE Hyperbola Previous Year Questions With Solutions: The word “hyperbola” comes from the Greek o, which means “overthrown” or “extreme,” and is also the source of the English term hyperbole. Menaechmus discovered hyperbolae while investigating the problem of doubling the cube, but they were initially referred to as sections of obtuse cones. In his definitive treatise on conic sections, the Conics, Apollonius of Perga (c. 262–c. 190 BC) is thought to have created hyperbola. The ellipse and parabola, the other two general conic sections, are named after the Greek words for “deficient” and “applied,” respectively; all three names are derived from Pythagorean terminology that related to a comparison of the side of fixed-area rectangles with a specified line segment.

Hyperbola Properties

1. On the hyperbola, the difference between the focal distances is constant, which is the same as the transverse axis ||PS – PS’|| = 2a.
2. The relationship e1-2 + e2-2 = 1 holds if e1 and e2 are the eccentricities of the hyperbola.
3. If the lengths of the transverse and conjugate axes are equal, a hyperbola is said to be rectangular or equilateral.
4. The eccentricity of a rectangular hyperbola is 2, the same as the length of the axes’ latus rectum.
5. If the point (x1, y1) is within, on, or outside of the hyperbola, the value of x12/a2 – y12/ b2 = 1 is positive, zero, or negative.
6. Two lines intersect the centre of the hyperbola. The asymptotes of the hyperbola are the tangents to the centre.
7. The hyperbola’s latus rectum can be described as a line perpendicular to the transverse axis that passes through either of the conjugate axis’ parallel foci. 2b2/a is the answer.

Hyperbola’s Important Characteristics

A hyperbola is a locus of a point in the plane with a constant ratio of the distance between the two fixed locations called foci. A mirror image of a parabola is known as a hyperbola. This article thoroughly explains the important properties of the hyperbola.

A hyperbola is generated when a right circular cone crosses a plane in such a way that the angle between the plane and the vertical axis is less than the vertical angle. The plane cuts the two nappes of the cone in hyperbola, resulting in the development of two discontinuous open curves.

The hyperbolic appearance of circles in conic sections is investigated.

Conic sections can be interpreted as a natural model of the geometry of perspective where the scene being viewed consists of circles, or more commonly, an ellipse, in addition to offering a uniform description of circles, ellipses, parabolas, and hyperbolas. The image of the scene is often a central projection onto an image plane, with all projection rays passing through a fixed point O, the centre, and the viewer is typically a camera or the human eye. The lens plane is parallel to the image plane at the lens O.

The image of a circle c is

1. A circle if circle c is in a particular position, such as parallel to the image plane and others (see stereographic projection),
2. An ellipse if c has no point in common with the lens plane,
3. A parabola if c has one point in common with the lens plane, and d) a hyperbola if c has two points in common with the lens plane.

Hyperbola Eccentricity

Eccentricity is a characteristic in conic sections that indicates how round they are. More eccentricity means less spherical behaviour, whereas less eccentricity means more spherical behaviour. It’s represented by the letter “e.”

The ratio of the distance between the focus and a point on the plane to the vertex and that point alone is the eccentricity of Hyperbola.

The distinction between a parabola and a hyperbola

Key Distinction:

A parabola is a conic section formed by cutting a conical surface parallel to the side of the cone with a plane. A hyperbola is formed when a plane slices a conical surface parallel to the axis.
In mathematics, parabola and hyperbola are separate words, sections, and equations describing two different portions of a cone. These differ in shape, size, and a variety of other aspects, including the methods used to calculate them. We must first comprehend the cone and its various conic portions to comprehend them.

Parabola

• A set of points in a plane equidistant from a straight line or directrix and focus is known as a parabola.
• The focus and directrix of a parabola are the same.
• e = 1 Eccentricity
• Regardless of size, all parabolas should have the same shape.
  A parabola’s two arms should be parallel to each other. It has no asymptotes.

Hyperbola

• The difference in distances between a set of points in a plane and two fixed points is a positive constant, according to the hyperbola.
• Two foci and two directrices make up a hyperbola.
• Eccentricity (e>1) is a term used to describe a person who is eccentric.
• Hyperbolas come in a variety of shapes.
• Hyperbola has arms that are not parallel to each other.
• There are two asymptotes.

FAQs

In geometry, what is a hyperbola?

A hyperbola is an open curve with two branches that are mirror reflections of one another. It's made up of two curves that resemble infinite bows. The hyperbola equation, foci, eccentricity, directrix, latus rectum, and properties of such curves will all be covered here.

What ideas will the new hyperbola section teach students?

Students will also learn about hyperbola equations under various conditions, normal and tangents of hyperbola, vertex, focii, eccentricity, axes, and hyperbola applications, among other things. Previous year's chapter-by-chapter solutions will aid students in better understanding ideas and question patterns.

Why should you practice JEE Math's previous year's questions?

Practicing JEE past year papers would aid JEE applicants in comprehending the question pattern, marking scheme, and level of complexity of the test paper. To aid JEE applicants in their preparation, we have created chapter-by-chapter past year questions with solutions, practice papers, and more.

What ideas will the new hyperbola section teach students?

Students will also learn about hyperbola equations under various conditions, normal and tangents of hyperbola, vertex, focii, eccentricity, axes, and hyperbola applications, among other things. Previous year's chapter-by-chapter solutions will aid students in better understanding ideas and question patterns.