Mathematics · Class 12 Science
Ch 15Hyperbola — Class 12 Mathematics, concept-first.
The previous chapter defined a conic's eccentricity and showed that gives a parabola and gives an ellipse. This chapter completes the family: a hyperbola is the case — the locus of a point whose distance from a fixed focus is always a constant multiple, greater than one, of its distance from a fixed directrix.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Hyperbola
A hyperbola is the conic obtained when the eccentricity exceeds 1 — the locus of a point whose distance from a fixed focus is always e times (e>1) its distance from a fixed directrix, giving a curve with two separate…
Most relevant Q&A
- If the angle between the asymptotes is $30^\circ$ then find its eccentricity of hyperbola.Preview
- Find the centre, foci, eccentricity, equation of the directrices, length of the latus rectum of the hyperbola. $x^2-4y^2=4$.Preview
- If the eccentricity of a hyperbola is $\dfrac{5}{4}$, then find the eccentricity of its conjugate hyperbola.Preview
- Find the equations of the tangents to the hyperbola $x^2 - 4y^2 = 4$ which are (i) parallel (ii) perpendicular to the line $x + 2y = 0$.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
The previous chapter defined a conic's eccentricity and showed that gives a parabola and gives an ellipse.
Standard Form of the Hyperbola
A hyperbola is defined the same way an ellipse is, except for one flipped inequality: it is the path traced by a point whose distance from a fixed point (the focus) stays a constant multiple of its di…
Eccentricity, Foci, Directrices and the Latus Rectum
Once and are fixed, every other measurement of the hyperbola follows automatically — none of them are independent choices.
Parametric Equations of the Hyperbola
Just as points on an ellipse can be written using a single angle parameter via its auxiliary circle, points on a hyperbola can be parametrised using the circle drawn on its transverse axis as diameter…
Tangent and Normal at a Point — Cartesian Form
The tangent and normal at a point on a hyperbola are built exactly the way they are for an ellipse or parabola — differentiate implicitly, or simply quote the standard result, which is easiest to reme…
Tangent and Normal at 'θ', and the Condition for Tangency
The same tangent and normal lines can also be written directly in terms of the parameter , without first computing the point's Cartesian coordinates — genuinely convenient when a problem is already ph…
Asymptotes of the Hyperbola
An asymptote of a curve is a straight line the curve gets arbitrarily close to without ever quite reaching, as you follow the curve out towards infinity — formally, a line is an asymptote of if as .
The Conjugate Hyperbola
Every hyperbola has a natural partner curve called its conjugate hyperbola: the one obtained by swapping the roles of the transverse and conjugate axes — i.e., the curve whose transverse axis is the o…
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1If the angle between the asymptotes is $30^\circ$ then find its eccentricity of hyperbola.Preview
- Q2Find the centre, foci, eccentricity, equation of the directrices, length of the latus rectum of the hyperbola. $x^2-4y^2=4$.Preview
- Q3If the eccentricity of a hyperbola is $\dfrac{5}{4}$, then find the eccentricity of its conjugate hyperbola.Preview
- Q4Find the equations of the tangents to the hyperbola $x^2 - 4y^2 = 4$ which are (i) parallel (ii) perpendicular to the line $x + 2y = 0$.Preview
- Q5Find the eccentricity of the hyperbola $16x^2-9y^2=144$.Preview
- Q6Find the equation of the tangent to the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ at the point $(a\sec\theta,\,b\tan\theta)$.Preview