Parabola Equations and Properties PDF

Conic Section: Parabola  Parabola -generated when the plane cuts parallel to one generator forming an unbounded curve projectiles, architecture, acoustics, generating energy or heat  Terms related to Parabola VERTEX Verte...

Conic Section: Parabola  Parabola -generated when the plane cuts parallel to one generator forming an unbounded curve projectiles, architecture, acoustics, generating energy or heat  Terms related to Parabola VERTEX Vertex (v) is the main point of the parabola which lies at the middle portion of the curve. It is also the midpoint of the fixed point of the curve and the given line.  Terms related to Parabola Focus (F) it is another important point of the parabola that is located inside of the curve. The focus is also known as the fixed point of the parabola.  Terms related to Parabola FOCAL DISTANCE(a) } or FOCAL LENGTH { FOCAL DISTANCE (a)- distance between focus to vertex -also equal to the distance between the vertex and directrix.  Terms related to Parabola Directrix line (D.L) is a line which is outside and parallel to the parabola curve. Terms related to Parabola LATUS RECTUM LATUS RECTUM (LR) – a chord that passes through the focus and is parallel to the directrix. The length of the latus rectum of a parabola is equal to 4 times its focal length. LR=4a Half of latus rectum is called semi-latus rectum. The end points of the latus rectum are defined as 𝑹𝟏 𝒂𝒏𝒅 𝑹𝟐. Terms related to Parabola AXIS OF PARABOLA AXIS OF PARABOLA- also known as axis of symmetry. -passes through vertex and focus -perpendicular to LR and directrix. Eccentricity (e) is the ratio of the distances from a fixed point to one of the points of the curve and from this point to the directrix line. Eccentricity value expresses the degree of roundness of the given curve. For a parabola, eccentricity value is always equal to 1. x General Equation of Parabola  Standard form of Equation of Parabola given Vertex (h,k) Standard form of Equation of Parabola if vertex is at (h,k) UPWARD RIGHT DOWNWARD LEFT PARABOLA Opening of the UPWARD DOWNWARD RIGHT LEFT curve Coordinates of (h,k) (h,k) (h,k) (h,k) the Vertex Coordinates of (h, k + a) (h, k - a) (h +a, k) (h – a, k) the Focus Equation of the y=k–a y=k+a x=h-a x=h+a Directrix Equation of the x=h x=h y=k y=k Axis Latus Rectum 4a 4a 4a 4a Endpoints of the (h+2a, k + a) (h+2a, k - a) (h+a, k +2a) (h-a, k +2a) Latus Rectum (h-2a, k + a) (h-2a, k - a) (h+a, k - 2a) (h-a, k - 2a) Find the equation of the parabola having vertex at (0,0), Focus at (2,0). Give the following : a) Orientation of parabola b) Focal Distance (a) c) length of Latus Rectum d) end points of the latus rectum e) equation of the directrix line f) Axis of the parabola g) Equation of parabola The equation of the directrix of the line of a parabola is x-5=0 and its focus is at (-5,0). Give the other analytical properties and the equation of the parabola a. Vertex b. Focal Distance(a) c. length of Latus Rectum d. end points of the latus rectum e. Axis of the parabola f. Equation parabola Find the properties of the parabola with equation 𝑦 2 + 4𝑥 = 0 a. Vertex b. Focus c. Axis of parabola d. Directrix Line equation e. Latus rectum f. endpoints of LR Formulate the equation of the parabola with vertex at (-5,6), the equation of its directrix line is x=-8 and opens to the right. 2. What is the equation of the parabola whose vertex and focus are located at (4,-1) and (4,-5) respectively ? Reduce the parabola equation into standard form 2 𝑥 + 3𝑦 − 2𝑥 − 8 = 0 Reduce the parabola equation into standard form 0 SEATWORK 1: Get the following parts of parabola: a: ______ Vertex: ________ Focus: ________ 0 LR: ________ Endpoints of LR: _____ Equation of DL: _____ Axis of Parabola: _____