7. (a) Find a centre of projection such that a given angle may be projected into any required ankle and a given line to infinity.

(b) Project a quadrilateral into a rectangle.

8. (a) Prove that anharmonic ratios are unchanged by projection.

(b) Deduce, by projection, any three properties of the parabola from the properties of the circle.

FINAL HONOUR MATHEMATICS.

Analytic Solid Geometry.

Note. Any seven questions.

1. Find the equations of a straight line through a given point and perpendicular to two given lines.

2. Explain what loci in space are determined by the following equations:-(a) f(x)=0; (b) f(x)=0, f(y)=0; (c) f(x, y)=0; (d) f(r)=0; (e)f(r,0)=0; (f)r=bcos@, (g) x2+y2=r2, z=a.

3. Find the equation of the plane through three given points; and interpret the meaning of it in terms of the projections of the given points on the coordinate planes.

4. (a) Transform the equation f(x, y, z)=0 from one set of rectangular axes to another rectangular set whose direction cosines are l1, m1, n1, l2, m, n, and l ̧, m3, n3. (b) Show that in (a) |l, m, n ̧|= ±1.

5. Find the equation of the tangent plane to the general cone with vertex at the origin.

(b) Deduce the tangential equation of the cone of (a). 6. For the system of conicoids

(a) any two intersect orthogonally at every point of

intersection;

(b) the conicoids are confocal;

(c) three real ones pass through any given point.

7. (a) Find the equation of a diametral plane of the general conicoid.

(b) Show that there are three, and only three, diametral planes that are perpendicular to the chords they bi

sect.

8. Find the equation of the curve of section of f(x, y, z)=0 and the plane lx+my+nz=p. Illustrate with the cone xy+ya+2x=0 and the plane r+y+2=4.

9. Obtain an expression for the radius of curvature of (a) a normal section, (b) an oblique section, of any surface in terms of the principal radii of curvature.

and illustrate by means of a graph.

(b) If f1(a) and f2(a) are both finite, prove that
L [f1(x) ƒ2(x)]=[L ƒ1(x)] [L ƒ2(x)].

and show that Rn can be expressed in either of the forms

(b) Show that the number 0 in the second remainder form in (a) approaches the limit 1/(n+2) as h approaches zero, provided f(n+2) (a)‡0,

4. (a) Derive the conditions for which the point (h, k) on the curve f(x, y)=0 is (1) a double point, (2) a cusp, (3) an isolated singular point.

(b) Find the singular point and asymptote of the curve 3x2-xy-2y2+-8y-0, and trace the curve.

5. Show the regions of the plane in which the curve lies which is represented by the equivalent equations

(x2+y2—1)2(x+y)(x−y)=2xy,
(x2+y2) (x2—y2) (x2+y2—2)

+[x−(1+V2)y] [x—(1—√2)y]=0.

Sketch the curve, making use of the principle of linear combination.

6. Derive Taylor's and Maclaurin's expansions for two variables.

7. (a) Derive the conditions for a maximum or minimum on a surface.

(b) Examine for maxima or minima

}y3—xy2+x2y—x=0.

8. (a) Under what conditions will the curves y=f(x) and y=0(x) have contact or order k?

(b) Find the equation of the conic having third order contact at the origin with the curve y=f(x).

9. Find the envelope of the family of ellipses whose axes coincide and whose area is constant.

10. (a) If F(xy)=0, prove that

(b) Find the partial derivatives and the total differential of