Question

Let o (77=27), + 27 x + 28 where = (1, x2) -Find the sysmetric matrix A with that Q7) AZ of Q (7)=1 is an elipse. Find the dogonal from of Git) it to detemine the semi-axis longth of the elipse The graph and

Answer 1

If you have any doubts in the solution please ask me in commment here i use matrix method to find all reqireds ...final answer length of axis i evaluate. Both length of major axis length and minor length

²= xzlazi 2 T solution: QIX) = 2x12 + 2X, X2 + 2x22 where to find Qă) = A sot. Νοω χΑχ. Let A = 412 = 921 A is symmetric au 21 aix, + 912 22 Tax= 1x, 22) (X, X2) A21x1 +92222 an a12 ) A21 A22 a 12 1(2) A22 2x2 2x) → TAX= 9,1x2 + 20122, X2 + 022x2² .:: 0 12 = 92)

Qã) - 2x² + 2x22, + 2x₂² By 0 & ② comparison ani = 2, 022 = 2, a 12 = 1, Azi = 1 2 1 A = [ 3 ] Answer 2 Now Qrã) = 1 is equation of ellipse Now to zind diagonal form of Qat) A = Let a be eigen value of a then 2 2-X 1 (A-11) = 0 (2-x)²-1=0 1 2-4 X = 1, 3, SO eigen value of o are A = 1,3 Now if a=1 (A-AI) (g) = 0 [ ][%]=[:] => 4, +4₂= -0 42 = -4, - [*]:15] - [1 Yg [-] if a = +3, (A-XI) y =0 [;'. .11% 1-10) -4, +42 Y = 42 [- [!] 9 vz = [ ] → → A is diagondiizable. A = PDP-1 Where D=[68]

Basis : NOW p = j ortho normal Now [-] ūs = 1 12 ſ = +182 -V2 > [ 1 [] es ETA Let where - z x= 5z. [3] XTA x = (ſz)T A CỔz) zTÃT A Řz. V V2 -1122 12 2 2 2 [2, 23] [1 II- - } [2, 31 [] [ 2+2] - 1/2 [2, 22] [ 32 ] T- Z2 +3ză 2.

> Q(x) = XTAX = > XT A.x 2,2+ 322² > Qix) = 1 2,2 + 3222 = 1 Answer > diagonal form of Qră) 2,2+ 32² 2,2+372. Now Qră) = 1 is ellipse =) 2,2 + 332 = 1 is ellipse > + Z₂² - 1 (112 ( )? minor exis = So Length (semi-axis) of ash } Ausuh . 1 semi-axis dength of major OxiS. =