Calendar: 1870-1871 Page 508

508 GENERAL LITERATURE AND SCIENCE Find also the length of the perpendicular upon the line from given point and show how the sign is regulated Find the condition that the two lines Ax By A'x B'y may be perpendicular to each other tbe axes being inclined at an angle ω Find the area of triangle in terms of the coordinates of its angular points and thence deduce the condition that three points whose coordinates are given may be in tbe same straight line Investigate formulae for passing from one system of coor- dinates to another with the same origin If IV -f- my' and px' -j qy' find the relation which must exist between If an expression of the form χ cos sin ρ be re- presented by determine tbe relation beiween the lines β β If ω be the angle between the first two of these lines show that the equation to the line through their intersection perpen- dicular to £ 0is -f£ cos ω cos ω Find the condition that the general equation of the second degree may represent two straight lines What further condition is required that the lines may be parallel If two triangles are co-axial prove that they are also co-polar and the converse Find the equation to the two straight lines which bisect the angles beiween the lines ax -f- hxy f- cy2 If the equation to conic be ax2 -j- hxy -j- by2 show that the equation to its axes will be xy χ1 י- 10 Find the ratio of the segments into which the line joining two given points is divided by given straight line Hence deduce the equation to the pair of tangents drawn to conic from an external point 11 In the general equation of the second degree with the con- stant term wanting show that the terms of the first degree equated to zero represent the tangent at the origin