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CULUS OF SEVERAL VARIABLES ‘ 691 6. Find the equations of the tangent plane and normal to the conicoid 7% ax? + by? bo cz? = 1 at (x4, Ya, 2a). 7, At what angle do the sphere x? + y? + z? = 16 and the plane y = 2 intersect? 2 y? 2 : x Show that the ellipsoid —- + +— + = land ' 8. 12 16 1 and the hyperboloid y? 22 = 1 inter 7 -x?-27 = 1 intersect orthogonally, 9, Show that the sphere x? + y? + 2? = 16 and the cone 3(x? + y#) = 2? intersect orhtogonally. 10. Find the equations of two tangent planes to 2x? ~ 6y? + 3z7 = 5 which pass through the line x + 9y - 3z = 0 = 3x - 3y-+ 62-5. 11. Prove that every normal to a sphere passes through the centre, State and prove its converse, ; “9, Prove that the ellipsoid 2x? + y? + z= 7 and cylinder y? = 4x are orthogonal-at (1, 2, 1). : ; 13, Determine a and b so as to make the paraboloid ,y = ax? + bz? orthogonal to the ellipsoid x? + y? + 22? = 7 at the point (1, 2, 1). : 12.6.5A EXTREMA OF FUNCTIONS OF TWO VARIABLES In chapter 5 we studied how to tind maximum and minimum values of a function of one variable. Now we will develop similar techniques ‘for functions of two variables. : ¥ + The graphs of many functions of two variables form hills and valleys. the tops of the hills are called relative maxima, and bottoms of the valleys are called: relative minima, ‘ , Geometrically, relative maxima-and minima are the high and low points in their immediate vicinity. : wn 12.6.5B DEFINITIONS Relative Maximum : ‘A function’ fof two variables is said to have a relative maximum at @ point (Xo. Yo) if there is ycirele centred ut (xX, Y,) such that fx. Yu) 2 tix. y) for all points (x, y) inside the circle... Relative Minimum A function f ef two variables is said t have a relative minimum at a point (X Yo) if there is a circle centred at (x,, Ya) Such that fO.. yd 7 HS. yd for all points (x. y) inside the circle. ” , Scanned with CamScanner 
