{"id":35229,"date":"2022-08-25T11:35:20","date_gmt":"2022-08-25T06:05:20","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35229"},"modified":"2022-08-25T11:35:20","modified_gmt":"2022-08-25T06:05:20","slug":"inter-2nd-year-maths-2b-hyperbola-formulas","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-2nd-year-maths-2b-hyperbola-formulas\/","title":{"rendered":"Inter 2nd Year Maths 2B Hyperbola Formulas"},"content":{"rendered":"

Use these Inter 2nd Year Maths 2B Formulas<\/a> PDF Chapter 5 Hyperbola to solve questions creatively.<\/p>\n

Intermediate 2nd Year Maths 2B Hyperbola Formulas<\/h2>\n

Definition:
\nA conic with an eccentricity greater than one is called a hyperbola, i.e., the locus of a point, the ratio of whose distances from a fixed point (focus) and fixed straight line (directrix) is e, where e> 1 is called a hyperbola.<\/p>\n

Equation of hyperbola in standard form:<\/p>\n

\u2192 \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1
\nHere b2<\/sup> = a2<\/sup> (e2<\/sup> – 1)<\/p>\n

\u2192 The difference of the focal distances of any point on the hyperbola is constant.
\n\"Inter
\n(i.e.,) SP ~ S’P = 2a<\/p>\n

\"Inter<\/p>\n

\u2192 A Point P(x1<\/sub>, y1<\/sub>) in the plane of the hyperbola S = 0 lies inside the hyperbola, if S11<\/sub> < 0, lies outside if S11<\/sub> > 0 and on the curve S11<\/sub> = 0<\/p>\n

\u2192 Ends of the latus rectum (\u00b1 ae, \u00b1 b2<\/sup>\/a) and the length of the latus rectum is 2b2<\/sup>\/a.<\/p>\n

\u2192 Equation of tangent at P(x1<\/sub>, y1<\/sub>) to \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1
\n\\(\\frac{x x_{1}}{a^{2}}-\\frac{y y_{1}}{b^{2}}\\) = 1<\/p>\n

\u2192 Equation of normal at P(x1<\/sub>, y1<\/sub>)
\n\\(\\frac{a^{2} x}{x_{1}}+\\frac{b^{2} y}{y_{1}}\\) = a2<\/sup> + b2<\/sup><\/p>\n

\u2192 Parametric form is x = a sec \u03b8, y = b tan \u03b8.<\/p>\n

\u2192 Equation of tangent at \u2018\u03b8\u2019 on the hyperbola
\n\\(\\frac{x}{a}\\) sec \u03b8 – \\(\\frac{y}{b}\\) tan \u03b8 = 1<\/p>\n

\u2192 Equation of normal at \u03b8 on the hyperbola
\n\\(\\frac{a x}{\\sec \\theta}+\\frac{b y}{\\tan \\theta}\\) = a2<\/sup> + b2<\/sup><\/p>\n

\u2192 A necessary and sufficient condition for a straight line y = mx + c to be tangent to hyperbola c2<\/sup> = a2<\/sup>m2<\/sup> – b2<\/sup>
\ny = mx \u00b1 \\(\\sqrt{a^{2} m^{2}-b^{2}}\\)<\/p>\n

\u2192 If P(x1<\/sub>, y1<\/sub>) is any point in the plane of the hyperbola S = 0 then the equation of the polar of P is S1<\/sub> = 0.<\/p>\n

\"Inter<\/p>\n

\u2192 The pole of the line lx + my + n = 0; n \u2260 0 with respect to the hyperbola S = 0 is
\n\\(\\left(\\frac{-a^{2} l}{n} ; \\frac{b^{2} m}{n}\\right)\\), n \u2260 0<\/p>\n

Equation of a Hyperbola in Standard From.<\/strong>
\nThe equation of a hyperbola in the standard form is \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1.
\nProof:
\nLet S be the focus, e be the eccentricity and L = 0 be the directrix of the hyperbola.
\nLet P be a point on the hyperbola. Let M, Z be the projections of P, S on the directrix L = 0 respectively. Let N be the projection of P on SZ. Since e > 1, we can divide SZ both internally and externally in the ratio e : 1.<\/p>\n

Let A, A’ be the points of division of SZ in the ratio e : 1 internally and externally respectively. Let AA’ = 2a. Let C be the midpoint of AA’. The points A, A’ lie on the hyperbola and \\(\\frac{\\mathrm{SA}}{\\mathrm{AZ}}\\) = e, \\(\\frac{\\mathrm{SA}^{\\prime}}{\\mathrm{A}^{\\prime} \\mathrm{Z}}\\) = e
\n\"Inter
\n\u2234 SA = eAZ, SA’ = eA’Z
\nNow SA + SA’ = eAZ + eA’Z
\n\u21d2 CS – CA + CS + CA’ = e(AZ + A’Z)
\n\u21d2 2CS = eAA’ (\u2235 CA = CA’)
\n\u21d2 2CS = e2a ^ CS = ae
\nAslo SA’ – SA = eA’Z – eAZ
\nAA’ = e(A’Z – AZ)
\n\u21d2 2a = e[CA’ + CZ – (CA – CZ)]
\n\u21d2 2a = e 2CZ (\u2235 CA = CA’) \u21d2 CZ = \\(\\frac{a}{e}\\).<\/p>\n

Take CS, the principal axis of the hyperbola as
\nx-axis and Cy perpendicular to CS as y-axis. Then S = (ae, 0).
\nLet P(x1<\/sub>, y1<\/sub>).
\nNow PM = NZ = CN – CZ = x1<\/sub> – \\(\\frac{a}{e}\\).
\nP lies on the hyperbola \u21d2 \\(\\frac{\\mathrm{PS}}{\\mathrm{PM}}\\) = e
\n\u21d2 PS = ePM \u21d2 PS2<\/sup> = e2<\/sup>PM2<\/sup>
\n\u21d2 (x1<\/sub> – ae)2<\/sup> + (y1<\/sub> – 0)2<\/sup> = e2<\/sup> \\(\\left(x_{1}-\\frac{a}{e}\\right)^{2}\\)
\n\u21d2 (x1<\/sub> – ae)2<\/sup> + y1<\/sub>2<\/sup> = (x1<\/sub>e – a)2<\/sup>
\n\u21d2 x1<\/sub>2<\/sup> + a2<\/sup> e2<\/sup> – 2x1<\/sub>ae + y1<\/sub>2<\/sup> = x1<\/sub>e2<\/sup> + a2<\/sup> – 2x1<\/sub>ae
\n\u21d2 x2<\/sup>(e2<\/sup> -1) – y2<\/sup> = a2<\/sup>(e2<\/sup> -1)
\n\u21d2 \\(\\frac{x_{1}^{2}}{a^{2}}-\\frac{y_{1}^{2}}{a^{2}\\left(e^{2}-1\\right)}\\) = 1 \u21d2 \\(\\frac{x_{1}^{2}}{a^{2}}-\\frac{y_{1}^{2}}{b^{2}}\\) = 1
\nWhere b2<\/sup> = a2<\/sup>(e2<\/sup> – 1)
\nThe locus of P is \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1.
\nThe equation of the hyperbola is \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1.
\nNature of the curve \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1<\/p>\n

Let C be the curve represented by \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1. Then
\n(i) (x,y) \u2208 C(x, -y) \u2208 C and (x, y) \u2208 C \u21d4 (-x,y) \u2208 C.
\nThus the curve is symmetric with respect to both the x-axis and the y-axis. Hence the coordinate axes are two axes of the hyperbola.<\/p>\n

(ii) (x, y) \u2208 C \u21d4 (-x, -y) \u2208 C.
\nThus the curve is symmetric about the origin O and hence O is the midpoint of every chord of the hyperbola through O. Therefore the origin is the center of the hyperbola.<\/p>\n

(iii) (x, y) \u2208 C and y = 0 \u21d2 x2<\/sup> = a2<\/sup> \u21d2 x = \u00b1a.
\nThus the curve meets x-axis (Principal axis) at two points A(a, 0), A'(-a, 0). Hence hyperbola has two vertices. The axis AA’ is called transverse axis. The length of transverse axis is AA’ = 2a.<\/p>\n

(iv) (x, y) \u2208 C and x = 0 \u21d2 y2<\/sup> = -b2<\/sup> \u21d2 y is imaginary.
\nThus the curve does not meet the y-axis. The points B(0, b), B'(0, -b) are two points on y-axis. The axis BB’ is called conjugate axis. BB’ = 2b is called the length of conjugate axis.<\/p>\n

(v) \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1 \u21d2 y = \\(\\frac{b}{a} \\sqrt{x^{2}-a^{2}}\\) \u21d2 y has no real value for -a < x < a.
\nThus the curve does not lie between x = -a and x = a.
\nFurther x \u2192 \u221e \u21d2 y ^ \u00b1 \u221e and
\nx \u2192 -\u221e \u21d2 y \u2192 \u00b1 \u221e.
\nThus the curve is not bounded (closed) on both the sides of the axes.<\/p>\n

(vi) The focus of the hyperbola is S(ae, 0). The image of S with respect to the conjugate axis is S'(-ae, 0). The point S’ is called second focus of the hyperbola.<\/p>\n

(vii) The directrix of the hyperbola is x = a\/e. The image of x = a\/e with respect to the conjugate axis is x = -a\/e. The line x = -a\/e is called second directrix of the hyperbola corresponding to the second focus S’.<\/p>\n

\"Inter<\/p>\n

Theorem:
\nThe length of the latus rectum of the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1 is \\(\\frac{2 b^{2}}{a}\\)
\nProof:
\nLet LL be the length of the latus rectum of the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1.
\n\"Inter
\nIf SL = 1, then L = (ae, 1)
\nL lies on the hyperbola \u21d2 \\(\\frac{(a \\mathrm{e})^{2}}{a^{2}}-\\frac{1^{2}}{b^{2}}\\) = 1
\n\u21d2 \\(\\frac{1^{2}}{\\mathrm{~b}^{2}}\\) = e2<\/sup> – 1 \u21d2 12<\/sup> = b2<\/sup>(e2<\/sup> – 1)
\n\u21d2 1 = b2<\/sup> \u00d7 \\(\\frac{\\mathrm{b}^{2}}{\\mathrm{a}^{2}}\\) \u21d2 1 = \\(\\frac{b^{2}}{a}\\) \u21d2 SL = \\(\\frac{b^{2}}{a}\\)
\n\u2234 LL’ = 2SL = \\(\\frac{2 b^{2}}{a}\\)<\/p>\n

Theorem:
\nThe difference of the focal distances of any point on the hyperbola is constant f P is appoint on the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) = 1 with foci S and S then |PS’- PS| = 2a
\nProof:
\nLet e be the eccentricity and L = 0, L’ = 0 be the directrices of the hyperbola.
\nLet C be the centre and A, A’ be the vertices of the hyperbola.
\n\u2234 AA’ = 2a
\n\"Inter
\nFoci of the hyperbola are S(ae, 0), S'(-ae, 0).
\nLet P(x1<\/sub>, y1<\/sub>) be a point on the hyperbola.
\nLet M, M’ be the projections of P on the directrices L = 0, L’ = 0 respectively.
\n\u2234 \\(\\frac{\\mathrm{SP}}{\\mathrm{PM}}\\) = e, \\(\\frac{\\mathrm{S}^{\\prime} \\mathrm{P}}{\\mathrm{PM}^{\\prime}}\\) = e<\/p>\n

Let Z, Z’ be the points of intersection of transverse axis with directrices.
\n\u2234 MM’ = ZZ’ = CZ + CZ’ = 2a\/e
\nPS’ – PS = ePM’ – ePM = e(PM’ – PM)
\n= e(MM’) = e(2a\/e) = 2a<\/p>\n

Notation: We use the following notation in this chapter.
\nS \u2261 \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) – 1, S1<\/sub> = \\(\\frac{x_{1}}{a^{2}}-\\frac{y_{1}}{b^{2}}\\) – 1
\nS11<\/sub> \u2261 S(x1, y1) = \\(\\frac{x_{1}^{2}}{a^{2}}-\\frac{y_{1}^{2}}{b^{2}}\\) – 1, S12<\/sub> = \\(\\frac{\\mathrm{x}_{1} \\mathrm{x}_{2}}{\\mathrm{a}^{2}}-\\frac{\\mathrm{y}_{1} \\mathrm{y}_{2}}{\\mathrm{~b}^{2}}\\) – 1<\/p>\n

Note: Let F(x1,<\/sub> y1<\/sub>) be a point and S \u2261 \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}\\) – 1 = 0 be a hyperbola. Then<\/p>\n