By Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)
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Additional resources for Analytical Geometry
Y\). The parametric equations are often useful in solving problems and we show their application in the following problems : 34 ANALYTICAL GEOMETRY Illustration I: Find the coordinates of the mirror image of (α, β) in the straight line ax+by+c = 0. The straight line through (α, β) perpendicular to ax+by+c = 0 has gradient b/a and so its parametric equations can be taken as x = a-\-t cos 0, y = ß+t sin 0 where tan 0 = b/a. Let the mirror image correspond to the value t of the parameter. The mid-point of the line joining (α, β) to its mirror image is (α+Jf cos 0, ß+\t sin 0).
Calculate the ratio ABI AC and explain the significance of the sign of the result. 63. The straight line given by x = t cos ψ—g, y = t sin ψ—/cuts the curve x2+y2+2gx+2fy+c = 0. Determine the values of t at the points of inter section and show that they are independent of ψ. Can you deduce anything about the curve from this result ? 64. Find the equation of the chord of the curve 3x2+4y2 = 28 whose mid point is the point (1, 1). Find also the length of this chord. ) 65. From the point P(l, 3) a line is drawn perpendicular to the line 8JC— 14y—31 = 0 to meet it in Q and PQ is produced to R so that PQ = QR.
It follows that Y χ' yjQ ο X Y' FIG. 17 x2+y2+2gx+2fy+c > 0 for all points outside the circle but < 0 for all points inside the circle. EXAMPLES 12. Which of the following points are inside the circle of radius 3 whose centre is at (1, - 2 ) : (i) (2, 3); (ii) (2, 2); (iii) (2, 1); (iv) ( 3 , - 1 ) ? 13. Obtain the lengths of the tangents from the origin to the circle = 0. x2+y2+7x-4y+16 14. Calculate the length of the tangents from (5, 12) to the circle x2 -f y2=69. 50 ANALYTICAL GEOMETRY 28. Circle with given diameter We now obtain the equation of the circle on the line joining the points A(*i, Ji) and A2(x2, y2) as diameter.
Analytical Geometry by Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)