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Quadratic Equations, , , , , , Quadratic Equation:, An equation of the form, ax? + bx +c = 0, where, , a#0, iscalleda, quadratic equation in x., , , , , , By Factorisation Method: In this, method, we first express the quadratic polynomial into product of linear, factors by using middle term splitting, method or different identities i.e., if ax?, , +bx +c can be factorized into two linear factors (x —a)(x—B), put (x—a) and, (x—B)=0 x-a=Oorx- B=Oie.,x, =aorx =, then a and f are the roots, of the equation ax? + bx +¢ =0., , , , , , , , , , , , , , , , , Roots of a Quadratic Equation:, Areal number «is called a root, of the quadratic equation ax? + bx, +c=0,a +0 ifaa’+ba+c=0., , Any quadratic equation can have at, , most two roots., , , , , , , , , , , Methods to, Find Roots, of Quadratic, Equation, , , , , , , , Quadratic Formula:, The roots of the quadratic, equation ax? + bx +¢ = 0,, a,b,c, € R anda #0 are, given by, , xn |b? — 4ac, , 2a, , , , , , , , Nature of Roots: For ax? + bx + c, a # 0 discriminant, (D), is given by b?— 4ac, , , , S. No. Discriminant Nature of Roots, D>0 Two distinct real roots, D=0 Two equal real roots, D<0 No real roots
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Arithmetic Progressions, , , , , , Sum of first 7 positive integers, LetS=14+24+3+..0, a=1,lastterm/=n, _n(a+l) _n(l+n), a a, , How many 2-digit numbers are, divisible by 3?, 2-digit numbers divisible by 3, are 12, 15, 18, ...99, a=12,d=3,4,=99, a,=a+(n-1)d, 99= 12+ (n—1)3, , te, pian as, 3, , , , , , , List of numbers in which each term is, obtained by adding a fixed number to the, preceding term except the first term. Fixed, number is called common difference., , a,a+d,a+2d,a + 3d,...a+(n-l) d, , , , , , , , , , , , , , , which provides the to and fro terms!, by adding subtracting from the present, , , , , , Ifa, b, c, are in AP, then, , bis arithmetic mean, , , , When first term and common, , difference are given :, S,= peat (nd), , a first term, d— common difference, n— total terms, , number., \« Can be positive or negative., , , , , , , , When first & last terms are given:), , é =F (ara,)ors, =Fa+)), , 1 last term, , , , a> first term d—> common difference, n— total terms qa, n® term, a> n term , 7 last term
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INTRODUCTION TO CIRCLES, , Circle and Line on a Plane: A Circle is a collection of all those points in a plane which are at a constant distance (radius) from a, fixed point (centre)., , © Fora circle and a line on a plane, there can be three possibilities:, (i) They can be non-intersecting., (ii) They can have a single common point i.e., the line touches the circle., (iii) They can have two common points i.e., the line cuts the circle., , OD, , When a line intersects the circles in such a way that there are two common points then that line is called secant., , SECANT, , Aue
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TANGENT, , © When the Line intersects the circle at only one point then, that line is known as a tangent., , P, A, {), , ¢ The tangent to a circle is a special case of secant, when the two end points of corresponding chord coincide., ¢ There is no tangent to a circle passing through point lying inside the circle., © There are exactly two tangents to a circle through a point outside the circle., , THEOREM 1, , ¢ The tangent at any point of a circle is perpendicular to the radius through the point of contact., The theorem states that “the tangent to the circle at any point is perpendicular to the radius of the circle that passes through, , the point of contact”., x 7 Q Y, , Here, O is the centre of the circle and OP | XY, Note:, , ¢ There is one and only one tangent to a circle passing through a point lying on the circle., * The line containing the radius through the point of contact is also called the ‘normal’ to the circle at the point., , NUMBER OF TANGENTS FROM A POINT ON A CIRCLE, , ¢ There is no tangent to a circle passing through a point lying inside the circle., , }, , © There is one and only one tangent to a circle passing through a point lying on the circle., , O
