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Hyperbola - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • Length of Latusrectum and Parametric Equation of Hyperbola, Conjugate Hyperbola is considered one of the most asked concept.

  • 113 Questions around this concept.

Solve by difficulty

The locus of the mid point of the chords of  \mathrm{\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1}, which pass through a fixed point \mathrm{P}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right) is

The equation of the hyperbola, in standard form, whose vertices are at ( \pm 5,0) and foci at ( \pm 7,0), is

The equation  \mathrm{\left|\sqrt{x^2+(y-1)^2}-\sqrt{x^2+(y+1)^2}\right|=K} will represents a hyperbola, if

A hyperbola has centre C and one focus at P(6, 8).  If its two directrices are \mathrm{3 x+4 y+10=0} and \mathrm{3 x+4 y-10=0} then CP=

If the equation \mathrm{4 x^2+k y^2=18}  represents a rectangular hyperbola, then k = 

 

 

If a hyperbola whose foci are S ≡ (2, 4) and S′ ≡ (8, –2) touches x-axis, then the equation of the hyperbola is

 

If hyperbola \mathrm{x^2-y^2=a^2}  and \mathrm{x y=c^2}  are of same size, then 

 

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The equation \mathrm{16 x^2-3 y^2-32 x-12 y-44=0} represents a hyperbola 

 

The vertices of the hyperbola \mathrm{9 x^2-16 y^2-36 x+96 y-252=0} are 

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If PQ is a double ordinate of the hyperbola \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1} such that CPQ is an equilateral triangle, C being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies

 

Concepts Covered - 3

What is Hyperbola?

Hyperbola:

A Hyperbola is the set of all points (x, y) in a plane such that the difference of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

OR,

The locus of a point which moves in a plane such that the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix) is constant. Constant is known as eccentricity e and for hyperbola e > 1.

 

Consider the figure, O is the origin, S and S' are the foci and ZM and Z'M' are the directrices.

The foci are S(ae, 0) and S'(-ae, 0). The equation of directrices: ZM is x = a/e and Z'M' is x = - a/e

P(x, y) is any point on the hyperbola and PM is perpendicular to directrix ZM.

\\ {\frac{\mathrm{PS}}{\mathrm{PM}}=\mathrm{e} \Rightarrow(\mathrm{PS})^{2}=\mathrm{e}^{2}(\mathrm{PM})^{2}} \\ {(\mathrm{x}-\mathrm{ae})^{2}+(\mathrm{y}-0)^{2}=\mathrm{e}^{2}\left(\mathrm{x}-\frac{\mathrm{a}}{\mathrm{e}}\right)^{2}} \\ {\mathrm{x}^{2}+\mathrm{a}^{2} \mathrm{e}^{2}-2 \mathrm{aex}+\mathrm{y}^{2}=\mathrm{e}^{2} \mathrm{x}^{2}-2 \mathrm{aex}+\mathrm{a}^{2}} \\ {\mathrm{x}^{2}\left(\mathrm{e}^{2}-1\right)-\mathrm{y}^{2}=\mathrm{a}^{2}\left(\mathrm{e}^{2}-1\right)} \\ {\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}\left(\mathrm{e}^{2}-1\right)}=1} \\ {\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}=1, \quad \mathrm{b}^{2}=\mathrm{a}^{2}\left(\mathrm{e}^{2}-1\right)}

The standard form of the equation of a hyperbola with center (0, 0) and foci lying on x-axis is

\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}=1 \quad \text { where, } \mathrm{b}^{2}=\mathrm{a}^{2}\left(\mathrm{e}^{2}-1\right)

 

Important Terms related to Hyperbola:

  • Centre: All chord passing through point O is bisected at point O. Here O is the origin, i.e. (0, 0). 

  • Foci: Point S and S’ is foci of the hyperbola where, S is (ae, 0) and S’ is (-ae, 0).

  • Directrices: The straight line ZM and Z’M’ are two directrices of the hyperbola and their equations are x = ae and x = -ae.

  • Axis: In figure AA’ is called the transverse axis and the line perpendicular to it through centre of hyperbola is called conjugate axis. 2a is length of transverse axis and 2b is length of conjugate axis.

  • Double Ordinate: If a line perpendicular to transverse axis of hyperbola meets the curve at Q and Q’, then QQ’ is called double ordinate. 

  • Latusrectum: Double ordinate passing through focus is called latus rectum. Here LL’ ans L1L1' are two latusrectum of a hyperbola.

 

Eccentricity of Hyperbola: 

\\\mathrm{Equation\;of\;the\;hyperbola\;is\;\;\;\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}\\\text{we have,}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;b^2=a^2\left ( e^2-1 \right )}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e^2=\frac{b^2+a^2}{a^2}}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e=\sqrt{1+\left ( \frac{b^2}{a^2} \right )}}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e=\sqrt{1+\left ( \frac{2b}{2a} \right )^2}}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;e=\sqrt{1+\left ( \frac{conjugate \;axis}{transverse \;axis} \right )^2}}

 

Focal Distance of a Point

The difference between the focal distance at any point of the hyperbola is constant and is equal to the length of the transverse axis of the hyperbola.

If P(x1, y1) is any point on the hyperbola.

\\\text{SP = ePM} = e\left ( x_1-\frac{a}{e} \right )=ex_1-a\\\\\text{S'P = eP'M} = e\left ( x_1+\frac{a}{e} \right )=ex_1+a\\\\\mathrm{|S'P-SP|=|ex_1+a-ex_1+a|=2a}

Length of Latusrectum and Parametric Equation of Hyperbola

Length of Latusrectum and Parametric Equation of Hyperbola: 

\\ {\text {Let Latusrectum } \mathrm{LL}^{\prime}=2 \alpha} \\ {\mathrm{S}(\mathrm{ae}, 0) \text { is focus, then } \mathrm{LS}=\mathrm{SL}^{\prime}=\alpha} \\ {\left.\left.\text {Coordinates of L and } \mathrm{L}^{\prime} \text { become (ae, } \alpha\right) \text { and (ae,} -\alpha\right) \text { respectively }} \\ {\text {Equation of hyperbola, } \quad \frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}=1} \\ {\therefore \frac{(\mathrm{ae})^{2}}{\mathrm{a}^{2}}-\frac{\alpha^{2}}{\mathrm{b}^{2}}=1 \Rightarrow \alpha^{2}=\mathrm{b}^{2}\left(\mathrm{e}^{2}-1\right)} \\\\\ {\alpha^{2}=\mathrm{b}^{2}\left(\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}\right) \quad\left[\mathrm{b}^{2}=\mathrm{a}^{2}\left(\mathrm{e}^{2}-1\right)\right]} \\ {\alpha=\frac{\mathrm{b}^{2}}{\mathrm{a}}} \\ {\Rightarrow 2 \alpha=\mathrm{LL}^{\prime}=\frac{2 \mathrm{b}^{2}}{\mathrm{a}}}

End-points of a latus rectum :

For LR passing through S(ae, 0):L\left(a e, \frac{b^{2}}{a}\right) ; L^{\prime}\left(a e,-\frac{b^{2}}{a}\right)

For LR passing through S'(-ae, 0):L_1\left(-a e, \frac{b^{2}}{a}\right) ; L_1^{\prime}\left(-a e,-\frac{b^{2}}{a}\right)

 

Parametric equation of Hyperbola:

The equations x = a secӨ, y = b tanӨ are called the parametric equation of the hyperbola

The circle with center O (0, 0) and OA as the radius is called the auxiliary circle of the hyperbola.

Draw PN perpendicular to the x-axis axis and NQ be a tangent to the auxiliary circle. Let be ∠QON = \theta (This angle is also known as Eccentric Angle).  Hence, the parametric equation of circle at point Q (a cosӨ, a sinӨ). 

\\ {\text {now, } \mathrm{x}=\frac{\mathrm{ON}}{\mathrm{OQ}} \cdot \mathrm{OQ}=\sec \theta \cdot \mathrm{a}} \\ {\mathrm{x}=\mathrm{a} \sec \theta} \\ {\mathrm{P}=(\mathrm{a} \sec \theta, \mathrm{y})} \\ {\mathrm{P} \text { lies on the hyperbola }} \\\\ {\frac{\mathrm{a}^{2} \sec ^{2} \theta}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}=1} \\\\\ {\mathrm{y}=\pm \mathrm{b} \tan \theta} \\\\ {\text { Point } \mathrm{P} \text { is }(\mathrm{a} \sec \theta, \mathrm{b} \tan \theta)}

Conjugate Hyperbola

Conjugate Hyperbola

Corresponding to every hyperbola, there exists a hyperbola in which the conjugate and the transverse axes of one is equal to the transverse and conjugate axes of the other. Such types of hyperbolas are known as the conjugate hyperbola.

\\ {\text {For the hyperbola }} \\\\ {\qquad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1} \\\\ {\text { The conjugate hyperbola is }} \\\\ {\qquad \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1}

 

 

 

 


 

Equation

\mathbf{\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1}

\mathbf{\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1}

Centre

(0 ,0)

(0, 0)

Vertices

(\pm \mathrm{a}, 0)

(0, \pm \mathrm{b})

Length of Transverse Axis

2a

2b

Length of conjugate Axis

2b

2a

Foci

(\pm \mathrm{ae},0)

(0, \pm \mathrm{be})

Distance b/w foci

2ae

2be

Equation of Directrices

\mathrm{x}=\pm \frac{\mathrm{a}}{\mathrm{e}}

\mathrm{y}=\pm \frac{\mathrm{b}}{\mathrm{e}}

Distance b/w Directrices

\frac{2 \mathrm{a}}{\mathrm{e}}

\frac{2 \mathrm{b}}{\mathrm{e}}

Eccentricity, e

\mathrm{e}=\sqrt{1+\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}}

\mathrm{e}=\sqrt{1+\frac{\mathrm{a}^{2}}{\mathrm{b}^{2}}}

Length of Latusrectum

\frac{2 b^{2}}{a}

\frac{2 a^{2}}{b}

Endpoint of Latusrectum

\left(\pm \mathrm{ae}, \pm \frac{\mathrm{b}^{2}}{\mathrm{a}}\right)

\left(\pm \frac{\mathrm{a}^{2}}{\mathrm{b}}, \pm \mathrm{be}\right)

Focal radii

|SP-S”P| = 2a

|SP-S”P| = 2b

Parametric coordinates

(\operatorname{asec} \theta, \text { b } \tan \theta), 0 \leq \theta \leq 2 \pi

(\operatorname{a\tan} \theta, \text { b } \sec \theta), 0 \leq \theta \leq 2 \pi

Tangent at vertices

x=\pm a

y=\pm b

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Shifted Hyperbola

The standard form of the equation of a hyperbola with center (h, k) and transverse axis parallel to the x-axis is

\frac{(\mathrm{x}-\mathrm{h})^{2}}{\mathrm{a}^{2}}-\frac{(y-k)^{2}}{b^{2}}=1

or

\frac{(\mathrm{X})^{2}}{\mathrm{a}^{2}}-\frac{(Y)^{2}}{b^{2}}=1

x = X + h    and y = Y + k

Origin is shifted to (h, k).

Study it with Videos

What is Hyperbola?
Length of Latusrectum and Parametric Equation of Hyperbola
Conjugate Hyperbola

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