Analytic Geometry Answers

Let u = (4,2,-1), v = (3, 1, 1) and w = (0, 2, 1). Compute the following:

(i) 2v - 3w -u

(ii) u(w+v)

(iii) ||u . w ||

(iv) the orthogonal projection of u on w

(v) the vector component of u orthogonal to w

Which of the following is an ellipse? More than one answer may be correct.

A. x^2÷3^2+y^2÷3^2=1

B. x^2=−y

C. y^2=4(1−x^2)

D. None of the above

Find the equation of the normal to the solid 2x^2 −y^2 +8z^2 = 11 at a point where it intersects the line x−3 = z =(y+1)/2.

A plane passes through (a,b, c) and cuts the axes in A,B,C, respectively, where

none of these points lie on the origin O. Show that the centre of the sphere OABC

satisfies the equation a/x +b/y +c/z = 2.

The normals at any point P of the ellipsoid x^2/9 +y^2+4 +z^2 = 1 meet the coordinate planes in Q1,Q2,Q3, respectively. Show that PQ1 : PQ2 : PQ3 :: 9 : 4 : 1.

Check whether the points (1,−1,−2),(1,−4,2),(3,0,2),(4,−3−2) are coplanar

or not. If they are coplanar, write the equation of the plane they pass through.

Otherwise, change the coordinates of one of the points so that they become

coplanar. In this case, find the plane passing through them.

Derive the equation (23) at page 42 of Unit 2, which represents the polar equation of a conic when the directrix L corresponding to a focus F is taken to the right of F.

Find the equation of the hyperbola with vertices (1,−4) and (1,4), and foci at(1,−6) and (1,6).

Give the equation of a conic which is symmetric to the line x+2 = 0.

Consider two lines L1 and L2 whose direction cosines l1,m1,n1 and l2,m2,n2 are given by the equations for l,m,n :

al +bm+cn = 0, f mn+gnl +hlm = 0,

where abc not = 0. Show that if L1 ⊥ L2, then f/a +g/b +h/c = 0.