{"id":3157,"date":"2023-10-04T12:11:43","date_gmt":"2023-10-04T12:11:43","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=3157"},"modified":"2023-10-04T12:23:00","modified_gmt":"2023-10-04T12:23:00","slug":"week-6-ss2-second-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-6-ss2-second-term-further-mathematics-notes\/","title":{"rendered":"Week 6 – SS2 Second Term Further Mathematics Notes"},"content":{"rendered":"

\u00a0WEEK SIX
\n<\/strong>CONIC SECTIONS: PARABOLA,ELLIPSE AND HYPERBOLA
\n<\/strong>THE PARABOLA
\n<\/strong>The parabola is a locus of points, equidistant from a given point, called the Focus<\/strong> and from a given line called the Directrix.<\/strong>
\n\t\t\"\"\/
\n\t\t(Length of directrix from V, (AV) = Length of Focus from V ,(FV))
\n<\/em><\/strong>The line AB, a distance ofa, <\/em><\/strong>from the y axis is called the Directrix.<\/strong> The line AF is called the axis of symmetry.
\n<\/em><\/strong>Since
\nBP = FP
\n\t\t\t<\/strong>BP2<\/sup> = FP2<\/sup>
\n\t\t(x + a)2<\/sup> = (x-a)2 <\/sup>+ (y-0)2
\n<\/sup>\u00a0\u00a0\u00a0\u00a0<\/strong>x2<\/sup> + 2ax + a2<\/sup> = x2<\/sup> – 2ax + a2 <\/sup>+ y2<\/sup>
\n\t\t\u00a0\u00a0\u00a0\u00a04ax = y2<\/sup>
\n\t\tthus, y2<\/sup> = 4ax <\/strong>is the equation of the parabola.
\nThe line RQ which is perpendicular to AF is called the latusrectum, V <\/strong>is called the vertex and F the focus<\/strong> of theparabola.
\nIf the vertex of the parabola is translated to a point \\(x1<\/sub>,y1<\/sub>), the equation of the parabola becomes
\n(y-y1<\/sub>)2<\/sup> = 4a(x-x1<\/sub>)2<\/sup>.
\n<\/strong>The above equation is said to be in the standard <\/strong>or canonical<\/strong>form
\nExamples
\n1. find the focus and directrix of the parabola y2<\/sup> = 16x
\n2. write down the equation of the parabola y2<\/sup>– 4y-12x+40 = 0 in its canonical form and hence find i) the vertex; ii) the focus; iii) the directrix of the parabola
\n `\u00a0\u00a0\u00a0\u00a0Solution<\/strong>
\n\t\t1. comparey2<\/sup> = 16x with y2<\/sup> = 4ax,
\n 4a = 16 , a = 4
\n Thus the focus<\/strong> is (4,0) while the directrix<\/strong> is x = -4
\n2. y2<\/sup>– 4y-12x + 40 = 0
\ny2<\/sup>– 4y-+4-12x+40 = 0+4 \u2026. (completing the square<\/em><\/strong>)
\ny2<\/sup>– 4y+4 = 12x – 36 \u2026\u2026 (rearranging<\/em><\/strong>)
\n (y – 2)2<\/sup> = 12(x-3) \u2026\u2026 (factorising<\/em><\/strong>)
\nBut (y \u2013 y1<\/sub>)2<\/sup> = 12(x-x1<\/sub>)
\n i) hence vertex (x1<\/sub>,y1<\/sub>) = (3,2)
\n ii) since 4a = 12, a = 3 then the focus (x1<\/sub>+a,y1<\/sub>) = (3+3,2) = (6,2)
\n iii)the equation of the directrix is x = 3-3 ie x=0
\nnote that the thedirectrix<\/strong> is of equal but opposite distance from the vertex<\/strong> with thefocus<\/strong>
\n\t\tthis means the distance between the focus<\/strong> and the vertex<\/strong> = the distance between the directrix<\/strong>and the vertex
\n<\/strong>
\n\u00a0
\n\u00a0Equation of Tangent and Normal at point (x1<\/sub>,y1<\/sub>) to a Parabola
\n<\/strong>1. Equation of Tangent
\n<\/strong> =
\n2. Equation of Tangent
\n<\/strong> = –
\nExample: find the equation of the tangent and normal to a parabola
\ni) y2<\/sup> = 12x at point (3,6)
\nii) y2<\/sup> = 16x at point (1, -4)<\/p>\n

\u00a0Solution
\n<\/strong>ii)y2<\/sup> = 16x , compare with y2<\/sup> = 4ax, thus a= 4<\/p>\n

\u00a0equation of tangent
\n<\/strong> = ‘ =
\nThus
\ny + 2x +2 =0 <\/strong> is the equation of the tangent<\/p>\n

\u00a0equation of the normal
\n<\/strong> = – , = –
\nThus
\n2y \u2013x + 9 = 0 <\/strong>is the equation of the normal
\n\t\t\t<\/strong>
\n\u00a0Evaluation
\n<\/strong>1. <\/strong>Find the foci and directrices of the following Parabolae
\n (a) y2<\/sup> = 32x (b) x2<\/sup> = 12y
\n2. Write the equation of the parabola, y2<\/sup> \u2013 6y \u2013 2x + 19 = 0 in the canonical form hence determine
\n Its vertex and focus<\/p>\n

\u00a0THE ELLIPSE
\n<\/strong>An ellipse <\/strong>is the locus of a point P, moving in a plane such that the sum of its distances from two fixed points F1<\/sub> and F2 <\/sub>called the foci<\/strong>, is a constant.
\n<\/strong>
\n\u00a0\"\"\/
\n\t\tOV = PF =a, OP = b and OF = c
\n<\/em><\/strong>Where V<\/em><\/strong> is the vertex<\/em><\/strong> or vertices<\/em><\/strong>, and F<\/em><\/strong> is the focus<\/em><\/strong> or foci<\/em> .
\n<\/strong>The equation of an ellipse is given by
\n + = 1 or + = 1 with centre (x1<\/sup><\/em> , y1<\/sup><\/em>) (a>b) major axis on x\u2026\u2026eqn(i)<\/em><\/strong><\/p>\n

\u00a0 + = 1 or + = 1 with centre (x1<\/sup><\/em> , y1<\/sup><\/em>) (a>b) major axis on y\u2026\u2026eqn(ii)<\/em><\/strong><\/p>\n

\u00a0a2<\/sup>= b2<\/sup> + c2
\n<\/sup><\/strong>wherea<\/em><\/strong> and b<\/em><\/strong> are on the major and minor axis respectively.
\n\"\"\/
\n\t\tExamples
\n1. Find the foci and four vertices of the ellipse + = 1
\n2. Write down the equation of the ellipse 25x2<\/sup>+ 4y2<\/sup>-50x-16y-59 = 0 in the canonical form and
\nhence find
\n i) the coordinates of the centre of the ellipse
\n ii) the four vertices of the ellipse
\n iii) the two foci of the ellipse<\/p>\n

\u00a0Solution
\n1. + = 1
\n Since a>b , then
\n + = + <\/p>\n

\u00a0By inspection,
\n<\/em><\/strong>a2<\/sup><\/strong>=25 and b2<\/sup>=9 thus a=+5 or -5 , b= +3 or -3 and c = + or –
\nc2<\/sup>=a2<\/sup>-b2<\/sup> , c= +4 0r -4
\ni) the foci f(0,c) = f1<\/sub>(0,4) and f2<\/sub>(0,-4)
\nii) the vertices V(a,0) = V1<\/sub>(3,0) and V2<\/sub>(-3,0)
\nthe co-vertices V(0,b) = V3<\/sub>(o,5) and V4<\/sub>(o,-5) <\/p>\n

\u00a02. \u00a0\u00a0\u00a0\u00a025x2<\/sup>+ 4y2<\/sup>-50x-16y-59 = 0
\n\t\t\t<\/em>\u00a0\u00a0\u00a0\u00a025x2<\/sup>-50x + 4y2<\/sup>-16y = 59
\n25(x2<\/sup>-2x +1-1)+ 4(y2<\/sup>-16y+4-4) = 59
\n25(x2<\/sup>-2x +1) -25+ 4(y2<\/sup>-16y+4)-16 = 59
\n 25(x2<\/sup>-2x +1) + 4(y2<\/sup>-4y+4) = 100
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a025(x-1)2<\/sup>+ 4(y-2)2 <\/sup> = 100
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 + = 1\u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 But + = 1
\n\u00a0\u00a0\u00a0\u00a0By inspection,<\/em><\/strong>a = +5 or -5 and b = +2 or -2
\n\t\t\t<\/strong> i) the coordinates of centre = (1,2)
\n ii) the vertices of the vertical axis are
\nV(0+x1<\/sub>,a+y1 <\/sub>) = V1<\/sub>(1,7)
\nV(0+x1<\/sub>,-a+y1 <\/sub>) = V2<\/sub>(1,-3)
\nvertices of the vertical axis are
\nV(b+x1 <\/sub>, 0 + y1<\/sub>) = V3<\/sub>(3,2)
\nV(-b+x1<\/sub> , 0+y1<\/sub>) = V4<\/sub>(-1,2)
\niii) the two foci are
\nF(0+x1<\/sub>,c+y1 <\/sub>) = F1<\/sub>(1 , +2)
\nF(0+x1<\/sub>,-c+y1 <\/sub>) = F1<\/sub>(1 , +2)<\/p>\n

\u00a0Equation of Tangent and Normal at (x1<\/sub> , y1<\/sub>) to an Ellipse
\n<\/strong> = is the equation of the tangent
\n = is the equation of the normal<\/p>\n

\u00a0Example: <\/strong>find the equation of the tangent and normal to the ellipse 4x2<\/sup> + 25y2<\/sup> = 100
\nEvaluation
\n1. Find the foci and vertices of the following ellipses
\n (a) 9x2<\/sup> + 10y2<\/sup> = 90 (b) 4y2<\/sup> + 5x2<\/sup> = 20
\n2. Write the equation of the ellipse, 4x2<\/sup> + 5y2<\/sup> – 24x \u2013 20y + 36 = 0 in the canonical form hence determine
\n Its vertices and foci <\/p>\n

\u00a0THE HYPERBOLA
\n<\/strong>The hyperbola is the locus of a point P<\/em><\/strong>, moving in a plane such that the distance from two fixed points called the foci <\/em><\/strong>have a constant difference<\/p>\n

\u00a0\"\"\/
\n\t\tThe equation of an ellipse is given by
\n<\/strong> – = 1 or – = 1 where = – <\/p>\n

\u00a0The equation of a Tangent to an ellipse at point (x1<\/sub> , y1<\/sub>) is given by
\n<\/strong> = <\/p>\n

\u00a0The equation of a Normal to an ellipse at point (x1<\/sub> , y1<\/sub>) is given by
\n<\/strong> =
\nExamples
\n1. Find the vertices and foci of the Hyperbola 25x2<\/sup> \u2013 16y2<\/sup> = 400
\n2. Find the equation of the tangent and normal to the Hyperbola 9x2<\/sup> \u2013 36y2<\/sup> = 36
\nSolution
\n<\/strong>1. 25x2<\/sup> \u2013 16y2<\/sup> = 400
\n – = 1 (in the canonical form)<\/em><\/strong>
\n\t\tSince – = 1 , then
\na = +4 or -4 b = +5 or -5 c = + or – hence
\nthe vertices are V1<\/sub>(4,0) and V2<\/sub>(-4,0)
\nthe foci are F() and F()<\/p>\n

\u00a0The General Conic
\n<\/strong>A conic<\/em><\/strong>in general may be defined as the locus of a moving point P, <\/em><\/strong>such that its distance fixed
\nPoint called the focus<\/em><\/strong>, and its distance from a fixed line called the directrix<\/em><\/strong> are in constant ratio.
\nThis constant ratio is called the eccentricity<\/em><\/strong> of the conic denoted by e<\/em><\/strong>. for <\/p>\n

    \n
  1. a parabola\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0e = 1\n<\/li>\n
  2. an ellipse\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0e < 1\n<\/li>\n
  3. a hyperbola\u00a0\u00a0\u00a0\u00a0e > 1\n<\/li>\n<\/ol>\n

    \u00a0GENERAL EVALUATION
    \n<\/strong>1. Write the equation of the ellipse, x2<\/sup> + 3y2<\/sup> + 2x \u201324y + 46 = 0 in the canonical form hence determine. Its vertices and foci
    \n2. Find the vertices and foci of the hyperbola 25x2<\/sup> \u2013 4y2<\/sup> = 100
    \n3. Find the equation of the tangent and normal to the parabola y2<\/sup> \u2013 18x = 0 at point (2,6)<\/p>\n

    \u00a0READING ASSIGNMENT: <\/strong>New Further Mathematics Project 3 by TuttuhAdegun and Godspower5th Edition\u00a0\u00a0\u00a0\u00a0
    \n<\/em>
    \n\u00a0WEEKEND ASSIGNMENT
    \n<\/strong>1. Find the equation of the tangent to the ellipse 4x2<\/sup> +9 y2<\/sup> = 36 at point (0,-2)
    \n A, 6y + x = 9 B, 3y + x = 9 C, 6y + 4x = 9 D, y +6 x = -9
    \n2. Find the foci of the ellipse 4x2<\/sup> + 9y2<\/sup> = 72
    \n A, 14 or -14 B, or – C, 5 or -5 D, or –
    \n3. Find the equation of the normal to the parabola y2<\/sup>-20x = 0 at (3 , 2 )
    \n A. 5y + x = 13 B. 3y + x = 9 C. 5y + x = 13 D. 5y + x = 13
    \n4. Which of the following is true eccentricity e <\/em><\/strong>of a parabola A.e< 1 B. e> 1 C. e = 1 D. e >= 1
    \n5. Which of the following is true eccentricity e <\/em><\/strong>of a hyperbola A. e < 1 B. e > 1 C. e = 1 D. e >=1<\/p>\n

    \u00a0THEORY
    \n<\/strong>1. (a) Show that the points Q(6 , 2) lies on a circle x2 <\/sup>+ y2<\/sup> \u2013 4x +2y -20 = 0 lies on a circle
    \n (b) Find the equation of the tangent to the circle at the point Q
    \n2. write the equation of the following ellipses in their canonical form and hence determine
    \n Their foci and vertices<\/p>\n

      \n
    1. 4x2<\/sup> + 5y2<\/sup> – 24x \u2013 20y + 36 = 0\n<\/li>\n
    2. 4x2<\/sup> + 6y2<\/sup> – 24x + 60y + 162 = 0\n<\/li>\n<\/ol>\n

      \t\t\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

      \u00a0WEEK SIX CONIC SECTIONS: PARABOLA,ELLIPSE AND HYPERBOLA THE PARABOLA The parabola is a locus of…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,253],"tags":[],"class_list":["post-3157","post","type-post","status-publish","format-standard","hentry","category-posts","category-second-term-ss2-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3157","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=3157"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3157\/revisions"}],"predecessor-version":[{"id":3158,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/3157\/revisions\/3158"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=3157"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=3157"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=3157"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}