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www.crackjee.xyz, A Resonance, Educating for better tomorrow, MATHEMATICS, Formula Book for, Engineering Entrance, Examinations, 13729, ISEET = IIT JEE + AIEEE + State PETS, (The Gateway to IITS, NITS, IITS, IISERS & Prominent Engineering/Science Institutions), Best Wishes for, Your Success in Competitive Examinations ahead !!!, This Booklet has been prepared by Resonance for the benefit & academic support to sincere students., You can send your feedback/suggestion on this booklet at [email protected]

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www.crackjee.xyz, Short Formula (Physics), Resonance, Educating for better tomorrow, 2, 9., A point and line:, ax, + by, + c, 1. Distance between point and line =, Va? +b?, X- X 1, y - y 1, ах, 2, + by 1+c, 1, 2. Reflection of a point about a line:, 2, a, a, X-X1 y-y1, b, ax1+by,+c, a2+b2, 3. Foot of the perpendicular from a point on the line is, a, a'x +b'y +c', Va?+b?, ax + by + c, 10., Bisectors of the angles between two lines:, Va?+b?, 2, a'2., a, b, c,, Condition of Concurrency :of three straight lines ax+ by + c, = 0, i = 1,2,3 is a, b, c2 = 0., a, b; c3, 11., 12., A Pair of straight lines through origin: ax? + 2hxy + by? = 0, 2yh?- ab, If e is the acute angle between the pair of straight lines, then tan 0 =, а +b, CIRCLE, 1., Intercepts made by Circle x? + y? + 2gx + 2fy + c = 0 on the Axes:, (b) 2 vf-c on y - aixs, (a) 2 Vg-c on x -axis, Parametric Equations of a Circle:, Tangent :, 2., x = h +r cos e; y = k + r sin 0, 3., (a) Slope form : y = mx ± a V1+m2, (b) Point form : xx, + yy, = a? or T = o, (c) Parametric form : x cos a + y sin a = a., 4., Pair of Tangents from a Point: ss,, = T2., 5., Length of a Tangent : Length of tangent is S,, 6., Director Circle: x2 + y2 2a? for x2 + y2 = a?, 7., Chord of Contact: T 0, 2 LR, 1. Length of chord of contact =, VR?+L?, RL, 2. Area of the triangle formed by the pair of the tangents & its chord of contact, R2+L?

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www.crackjee.xyz, Short Formula (Physics), V Resonance, Educating for better tomorrow, 3, 2RL, 3. Tangent of the angle between the pair of tangents from (x,, y,) =2 p2, 4. Equation of the circle circumscribing the triangle PT, T, is : (x - x,) (x + g) + (y - y,) (y + f) = 0., Condition of orthogonality of Two Circles: 2 g,9, + 2 f, f, = c, + c,., Radical Axis :S, - s, = 0 i.e. 2 (g,- 9,) x + 2 (f, - f,) y + (c,- c,) = 0., Family of Circles: s, + K S, = 0, S + KL = 0., 8., 9., 10., PARABOLA, Equation of standard parabola :, y? = 4ax, Vertex is (0, 0), focus is (a, 0), Directrix is x + a = 0 and Axis is y 0, Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a) & L' (a, - 2a)., 1., 2., Parametric Representation: x = at? & y = 2at, 3., Tangents to the Parabola y? = 4ax:, 1. Slope form y = mx +, a, (m + 0), 2. Parametric form ty = x + at?, m, 3. Point form T = 0, 4., Normals to the parabola y? = 4ax :, %3D, (x – x,) at (x, y,) ; y = mx – 2am – am³ at (am?, – 2am) ; y + tx = 2at + at at (at?, 2at)., 2a, y - y, =-, ELLIPSE, x2, Standard Equation :, a2, 1., = 1, where a > b & b? = a? (1 – e?)., b2, b2, (0 < e < 1),, a2, a, Directrices : x = +, e, Eccentricity: e =1, Focii : S= (+ a e, 0). Length of, major axes = 2a and minor axes = 2b, Vertices : A' = (- a, 0) & A = (a, 0)., 26° -2a(1-e?), Latus Rectum :=, a, Auxiliary Circle : x? + y2 = a?, Parametric Representation : x = a cos 0 & y b sin 0, Position of a Point w.r.t. an Ellipse:, 2., 3., 4., -1 > < or = 0., The point P(x, y,) lies outside, inside or on the ellipse according as ;, a2, 5., Line and an Ellipse: The line y = mx + c meets the ellipse, x2, = 1 in two points real, coincident, a?, b2, or imaginary according as c? is < = or > a?m? + b?., XX1 yy1, 6., Tangents: Slope form: y = mx ±, 2m²+b² , Point form :, =D1, a, b', Parametric form:, xcose ysine, =1, a

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www.crackjee.xyz, Short Formula (Physics), A Resonance, Educating for better tomorrow, 4, a'x_b?y, 7., Normals:, X1, = a? – b?, ax. sec 0 – by cosec 0 = (a? – b?), y = mx –, y1, h2m2, va+b-m?, 8., Director Circle: x? + y? = a? + b?, HYPERBOLA, x2, Standard Equation: Standard equation of the hyperbola is, a?, y2, = 1, where b2 = a? (e? – 1)., b2, 1., Focii : S = (± ae, 0) Directrices : x = +, Vertices : A = (t a, 0), 262, = 2a (e? – 1)., a, Latus Rectum (e): ( =, x², Conjugate Hyperbola :, a?, y², = 1, b?, 2., &, = 1 are conjugate hyperbolas of each., +, a, b?, Auxiliary Circle : x? + y? = a?., Parametric Representation : x = a sec 0 & y b tan e, Position of A Point 'P' w.r.t. A Hyperbola :, 3., 4., 5., S, =, a?, -1 >, = or < 0 according as the point (x, y,) lies inside, on or outside the curve., b2, 6., Tangents :, (i), Slope Form : y = m x±Va’m? - b?, X 1, Point Form : at the point (x, y,) is, 2, a, yy1, = 1., b', (ii), (iii), x sece, y tan 0, = 1., Parametric Form :, b, 7., Normals :, ax, at the point P (x,, y,) is, X1, b²y, = a? + b? = a?e?., (a), y1, ах, by, +, sece, tane, (b), at the point P (a sec 0, b tan 0) is, = a? + b? = a? e?., a+b2, (c), Equation of normals in terms of its slope 'm' are y = mx ±, Va--b-m, X, y, = 0, X, y, = 0. Pair of asymptotes:, b, 8., Asymptotes :, and, = 0., a, b, a