Analytical Geometry by Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon

By Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

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Obtain the coordinates of the centre and the radius of the circle repre­ sented by 3x2+3y2-6x+4y-l = 0. 3. Obtain the equation of the circle through the three points (1, 3), (2, — 1) and (-1,1). 4. Find the equation of the diameter of the circle x2-\-y2—2x+4y = 0 which passes through the origin. 5. Find the point which is diametrically opposite to (2, 1) on the circle x2+y2-ix+5y-4 = 0. 6. Prove that the points (9, 7) and (11, 3) lie on a circle with the origin as centre. Determine the equation of the circle.

30. A triangle is formed by the three lines x+y = 1, 3x—y = 7 and 3y=x+3. Calculate (a) the area of the triangle, (b) the angles of the triangle, (c) the coordinates of the circumcentre of the triangle. ) CHAPTER III Straight Lines 21. Homogeneous equation of the second degree The general homogeneous equation of the second degree can be written ax2+2hxy+by2 = 0. That is, (ax+hyf - (h2-ab)y2 = 0, and so the equation represents the two straight lines ax+{ h+V(h2-ab) }y = 0 and }y = 0 ax+ { h-V(h2-ab) through the origin.

45. Find the equation of the tangents from (2, —3) to the circle JC 2 +^ 2 + 6 J C - 4 ^ + 8 = 0. 46. Prove that the tangents from the origin to the circle x2 +j>2-\-2gx+2fy + c = 0 are mutually perpendicular if g2 +f2 = 2c. 35. Parametric treatment of the circle Consider the point P on the circle of radius a, centre the origin, and let the angle which OP (Fig. 23) makes with OX be Θ. Then Y χ' 1 o 1 x Υ' FIG. 23 the coordinates of P are (a cos 0, a sin 0), and so the circle may be represented by the parametric equations x = a cos Θ, y = a sin 0, where Θ is a parameter.

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