3. TRANSLATED PARABOLAS

Transcription

1 3. TRANSLATED PARABOLAS The Parabola with Verte V(h, k) and Aes Parallel to the ais Consider the concave up parabola with verte V(h, k) shown below. This parabola is obtained b translating the parabola 2 = 4a h units horizontall and k units verticall. Ais of smmetr ( h) 2 = 4a( k) k a a h S (h, k + a) V (h, k) = k a The verte V(h, k) lies midwa between the focus and the directri. The focal length is a. State the coordinates of the focus and the equation of the directri. [42].. We can now re-write the equation with reference to the aes. The equation is ( h) 2 = 4a( k). Copright MATRIX EDUCATION 2017 Page 265 of 289 Our Students Come First!

2 The equation of a parabola with focus S(h, k + a) and directri = k a is ( h) 2 = 4a( k). ( h) 2 = 4a( k) Equation of a concave up parabola with V(h, k) We can also conclude that the equation of a parabola with focus S(h, k a) and directri = k + a is ( h) 2 = 4a( k). This is shown in the diagram below. ( h) 2 = 4a( k) Equation of a concave down parabola with V(h, k) Ais of smmetr = k + a a k V (h, k) a h S (h, k a) ( h) 2 = 4a( k) Copright MATRIX EDUCATION 2017 Page 266 of 289 Our Students Come First!

3 Eample: Find the equation of parabola = 0 and sketch, clearl identifing the focus, verte and directri. Step 1: Complete the square, and epress in the form ( h) 2 = 4a( k). [43] Step 2: Identif the focal length. a =... [44] Step 3: Hence, determine the verte, focus and directri. [45] Step 4: Sketch the locus, showing all essential features. O Copright MATRIX EDUCATION 2017 Page 267 of 289 Our Students Come First!

4 (a) On the number plane below, plot the point S(2, 2) and draw the line = (b) Find the locus of a point which moves in a plane so that its distance from the point S(2, 2) is equal to its distance from the line = 4. [46] 2 Note to Students: In this question, ou are required to use the definition of a locus of a parabola to determine the equation of a parabola. Copright MATRIX EDUCATION 2017 Page 268 of 289 Our Students Come First!

5 Find the equation of the parabola with the following features: [47] (a) Focus (2, 1) and directri = [48] (b) Verte (2, 2), focal length 2 units, ais of smmetr = 2 2 [49] (c) Focus (2, 1) and verte (2, 2) 2... [50] (d) Focus (1, 7) and directri = 3 2 (e) Verte (0, 3 2 ), directri = 1 2 [51] 2... Copright MATRIX EDUCATION 2017 Page 269 of 289 Our Students Come First!

6 (f) Verte ( 2, 3), with the ais of smmetr parallel to the -ais, passing through the point ( 6, 5). [52] 2 Note to Students: Before determining the equation of the parabola, solve for the value of a.... Copright MATRIX EDUCATION 2017 Page 270 of 289 Our Students Come First!

7 A parabola has equation 2 = 8(4 ). (a) Draw a neat sketch of the parabola and clearl indicate on it the equation of its directri, the coordinates of its focus and the coordinates of all points of intersection of the parabola with the coordinate aes. 1 0 (b) Another parabola with equation 2 = 8 cuts the parabola 2 = 8(4 ) at A and B. Find the coordinates of A and B. [53] 2 Copright MATRIX EDUCATION 2017 Page 271 of 289 Our Students Come First!

8 A parabola has equation = (a) Find the coordinates of its verte and focus. [54] 2 Note to Students: Epress in the form ( h) 2 = 4a( k) b completing the square. [55] (b) Find the equations of its directri and latus rectum. 1 (c) Sketch the parabola, showing all essential features. 1 0 Copright MATRIX EDUCATION 2017 Page 272 of 289 Our Students Come First!