Pre-Calculus Review PDF

PRE-CALCULUS Mr. Aaron James D. Sebastian It is defined as a set of points in a plane whose sum of distances from two fixed points 𝐹1 and 𝐹2 is constant. ELLIPSE What do we call to the distance between the center and the focus? FOCAL DISTANCE Given th...

PRE-CALCULUS Mr. Aaron James D. Sebastian It is defined as a set of points in a plane whose sum of distances from two fixed points 𝐹1 and 𝐹2 is constant. ELLIPSE What do we call to the distance between the center and the focus? FOCAL DISTANCE Given the point P connecting to the two fixed points, to what part of an ellipse is equal to the sum of the distances from point P to the two fixed points? 𝑴𝒂𝒋𝒐𝒓 𝑨𝒙𝒊𝒔  It is the line passing through the vertices, foci and of an ellipse. Principal Axis These are the two points on the ellipse that lie on the principal axis. Vertices Determine the focal distance if the major axis and minor axis are 26 units and 10 units respectively. 𝟏𝟐 𝒖𝒏𝒊𝒕𝒔  It is a line segment that is perpendicular to major axis and connects the co-vertices of an ellipse Minor Axis What are the two standard form of equations of an ellipse with center at the origin? 𝟐 𝟐 𝟐 𝟐 𝒙 𝒚 𝒙 𝒚 𝟐 + 𝟐 = 𝟏 𝟐 + 𝟐 = 𝟏 𝒂 𝒃 𝒃 𝒂 If the center of an ellipse is at the origin and the foci of an ellipse is at (0,8) and (0,-8), what will be the form of equation of an ellipse? 𝟐 𝟐 𝒙 𝒚 𝟐 + 𝟐 = 𝟏 𝒃 𝒂 Find the equation of an ellipse, if the center of an ellipse is at the origin and major axis is vertical and 26 units and minor axis is 10 units. 𝟐 𝟐 𝒙 𝒚 + =𝟏 𝟐𝟓 𝟏𝟔𝟗  TRUE OR FALSE The value of a is always greater than the value of b and c. TRUE If the values of a and c are given, what formula are you going to use to find the value of b? 𝟐 𝟐 𝟐 𝒂 =𝒃 +𝒄 𝟐 𝟐 𝟐 𝒃 =𝒂 −𝒄 Find the equation of an ellipse, if the center of an ellipse is at the origin vertices at 0, ±10 and co-vertices at ±8,0 𝟐 𝟐 𝒙 𝒚 + =𝟏 𝟔𝟒 𝟏𝟎𝟎 Find the equation of an ellipse, if the center of an ellipse is at −1,1 , horizontal major axis, focal distance of 20 units and minor axis of length 42 units. (𝒙 + 𝟏)𝟐 (𝒚 − 𝟏)𝟐 + =𝟏 𝟖𝟒𝟏 𝟒𝟒𝟏  Determine if the given equation is an equation of an 𝑥−2 2 y2 ellipse, − =1 16 25 𝑵𝒐, 𝒔𝒊𝒏𝒄𝒆 𝒕𝒉𝒆 𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒊𝒔 𝒔𝒖𝒃𝒕𝒓𝒂𝒄𝒕𝒊𝒐𝒏  Determine if the given equation is an equation of an 𝑥−1 2 2 ellipse, + 𝑦+6 =1 36 𝒀𝒆𝒔  Determine the orientation of the major axis given the equation 𝑥 2 𝑦 2 + =1 36 49 Vertical, since the denominator of the second term is greater than the first term. Determine focal distance given the equation of an ellipse, 𝑥 2 𝑦 2 + =1 36 100 8 units, since a = 10 and b = 6 Complete the irony. CIRCLE : A = B ELLIPSE : 𝑨____ ≠𝑩 What are the two standard form of equations of an ellipse with center at the (h,k)? 𝒉)𝟐 𝒌)𝟐 𝟐 𝟐 (𝒙 − 𝒉) (𝒚 − 𝒌) (𝒙 − 𝒉) (𝒚 − 𝒌) 𝟐 + 𝟐 = 𝟏 𝟐 + 𝟐 = 𝟏 𝒂 𝒃 𝒃 𝒂  2 2 Convert 3𝑥 + 4𝑦 − 24 = 0 into the standard form. 𝟐 𝟐 𝒙 𝒚 + =𝟏 𝟖 𝟔  2 2 Convert 5𝑥 + 7𝑦 − 35 = 0 into the standard form. 𝟐 𝟐 𝒙 𝒚 + =𝟏 𝟕 𝟓 What are the important points needed in graphing an ellipse? Center, Foci, Vertices, and Co-vertices  TRUE OR FALSE The distance between two vertices is the length of major axis. TRUE  𝑥 2 𝑦 2 Convert + = 1 into the 6 7 general form. 𝟐 𝟐 𝟕𝒙 + 𝟔𝒚 − 𝟒𝟐 = 𝟎  𝑥 2 𝑦 2 Convert + = 1 into the 12 6 general form. 𝟐 𝟐 𝒙 + 𝟐𝒚 − 𝟏𝟐 = 𝟎 In In graphing an ellipse if the major axis is horizontal, where are you going to plot the vertices from the center? 𝑹𝒊𝒈𝒉𝒕 𝒂𝒏𝒅 𝑳𝒆𝒇𝒕 In In graphing an ellipse if the major axis is vertical, where are you going to plot the co-vertices from the center? 𝑹𝒊𝒈𝒉𝒕 𝒂𝒏𝒅 𝑳𝒆𝒇𝒕 CLOSING PRAYER Act of Contrition O my God, I am heartily sorry for having offended Thee, and I detest all my sins, because I dread the loss of heaven, and the pains of hell; but most of all because they offend Thee, my God, Who are all good and deserving of all my love. I firmly resolve, with the help of Thy grace, to confess my sins, to do penance, and to amend my life. Amen.