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8. Apply the method of coordinates to determine the side of a square inscribed in a triangle, taking for axes the base and perpendicular to base through vertex.
9. Determine the equation of a circle passing through the point x = 1, y = 2, and through the intersections with the axes of the line x - Y +3=0. Find also the equation of the second line in which this circle meets the
shall have a common root may be written in the form
(ac – 62) (a'c' – b^2) = (ac' + ca' – 266°)2. How are the roots of the quadratics related if the right-hand member of this equation vanish ?
12. Solve the simultaneous equations
x2 + 3x + y = 7,
y2 + x – 3y = 4.
NR. W. R. ROBERTS.
13. Find the equation of the circle passing through the intersection of the circle
x2 + y2 + 6x + 5y + 4 and the line 3x + y = 0, and also passing through the point 1, 0.
14. Find the angle between the tangents from the point 12, I to the circle
x2 + y2 - 4x – 2y – 20 = 0. 15. Find the values of 1 so that the line joining the points
41, 31 + 1; 1+1, 21 + 1, may be cut harmonically by the cirle
x2 + y2 – 20 = o.
16. Find the equation of the circle described on the chord of intersection of the circles
x2 + y2 - 25 = 0, x2 + yö + 3x + 4y – 24 = 0, as diameter.
17. If (Q1 + V - I b) (az + V – I bz) (az + V – 1 63) = P+V-IQ, find P2 + Q2.
18. If tan A = V5 – 2, tan B = V10 – 3, find tan 2 (A + B).
1. Find the condition that two of the lines represented by the equation
ax3 + 3bx? y + 3cxyz + dy3 = 0 should be at right angles.
2. Taking as triangle of reference the triangle formed by joining the centres of three given circles, tind in trilinear coordinates the equations of the four axes of similitude of these circles.
3. A parabola being traced on the paper, find the position of its axis and focus.
4. Solve the following equations :
5. Find a superior limit, as low as possible, of the positive roots of the equation
24 – 2x3 – 16.x2 – 513 – 600 = 0.
MR. F. PURSER.
6. Show that the equation
22 cos 20 + 2xy cos (0 - 0) + y cos 20 + 2x cos 6 + 2y cos 0+1=0 represents two right lines, and determine their equations.
7. Determine the trilinear coordinates of the centre of the circle passing through the feet of perpendiculars from the vertices of the triangle of reference on the opposite sides, and verify that this point lies in directum with the orthocentre and centre of circumscribing circle of the triangle.
8. If the three coaxal circles
2 + y2 + 2kx + 82 = 0,
22 + y2 + 2 k'a + 82 = 0,
202 + y2 + 2k" x + 82 = 0,
are cut by another circle at the angles 0, o', o”, prove that
where r, m', qul are the radii of the three circles.
9. Calculate for the cubic 23 + p.x2 + 2x + r the symmetric function
10. Prove the following rule with respect to the sign of any first minor of a determinant:
The sign of minor corresponding to any constituent is the same or opposite to that of the determinant obtained by cancelling corresponding row and column in situ, according as the constituent is at an even or odd interval from the diagonal line from the left-hand top to the right-hand bottom corner.
MR. W. R. ROBERTS.
11. Solve the equation
(1 + x)4 = a (1 + x4). 12. Given base and sum of sides of a spherical triangle, prove that the product of the sines of the perpendiculars from the extremities of the base on the external bisector of the vertical angle is constant.
13. Invert the following theorem :
Given base and sum of squares of sides, the locus of the vertex is a circle.
14. Find the locus of a point such, that if it be joined to the vertices of a triangle, and perpendiculars to the joining lines erected at the vertices, these perpendiculars meet in a point.
15. Find the value of the determinant
sin B sing
1. Prove the determinant relation connecting the cosines of the six arcs joining four points on the surface of a sphere, and derive from it the corresponding relation in the plane.
2. If a, B, , 8, and á', B', 7, 8' be the roots of the two biquadrati equations
20.24 + 40123 + 642x2 + 4238 + 24 = 0,
d'oct + 40'123 + 6d 2 x2 + 4a' 3x + d'4 = 0; prove that the value in terms of the coefficients of the determinant
(a-a) (a – B)* (a - 0) (a - 8)
contains the factor
ao a's - 401d's + baza'2 – 4a3a'1 + 24 d'o. 3. Find an expression for the area of a triangle formed by three points whose trilinear coordinates are given.
4. Apply Bezout's method to express in the form of a determinant the result of eliminating x from two given biquadratic equations; and show in general that the resultant so obtained contains no extraneous factor. 5. Given the equations of two circles
(x -a)? + (y - 3)2 - y2 = 0, (+-a')' + (y - B')? 72 = 0 : if tangents PT, PT', drawn at points T and T", one on each circle, be at right angles; find the equation of the locus of the middle point of the line TT', and give its geometrical interpretation.
MR. F. PURSER.
6. If t denote the length between points of contact of either external common tangent to two circles S, S', give the geometrical signification of the following equations :
S - $' - t2 = 0,
S' - S - t* = 0, (2) S + $' – t2 = 0,
V S + V8' + t = 0.
7. If a tangent to a central conic meet two parallel tangents, prove that the rectangle under its segments is equal to the square of the paralel semidiameter.
8. One leg of a right angle is of given length, and its extremity rests on a given line, while the other leg passes through a fixed point. Determine the equation of the locus described by the vertex.
9. State and prove Sturm's theorem for the case of an equation which has no equal roots.
10. If 1, M, v denote the roots of Euler's reducing cubic for a biquadratic, prove that
3H ^ + M + v=
and hence show that when the roots are all real, H and 2HI – 30J are both negative.
MR. W. R. ROBERTS.
11. Show how to describe a circle touching three given circles. 12. Find the centres of similitude of the circles
find the equation whose roots are
(a – B) (a - y), (B - y) (B - a), (9- «) (y – B). 14. Prove that if a determinant vanish, its minors A1, A2, &c., are respectively proportional to B1, B2, &c.
15. Prove that the sum of the squares of the reciprocals of any two diameters of an ellipse at right angles is constant.