. Algebraic geometry; a new treatise on analytical conic sections . Fig. 84. ^ n • Aa;+B« + Cby definition of the parabola. Aa; + By + C AKT. 136.] THE PAEABOLA. 123 Multiplying up and squaring, we have (A2 + B2) [{X - x^Y + {y- y^f] = (Aa; + By + 0)^,the required Corollary. Transposing all terms to the left, the terms ofthe second degree = x\^^ + &- A^) - 2^Bxy + y%A^ + B^ - B^)= B^x^-2fiiBxy + f<Y= (Bx- kyy, a perfect we see that in the equation of any parabola, the termsof the second degree form a perfect square. This is the distinguishing characteristic of the equ
. Algebraic geometry; a new treatise on analytical conic sections . Fig. 84. ^ n • Aa;+B« + Cby definition of the parabola. Aa; + By + C AKT. 136.] THE PAEABOLA. 123 Multiplying up and squaring, we have (A2 + B2) [{X - x^Y + {y- y^f] = (Aa; + By + 0)^,the required Corollary. Transposing all terms to the left, the terms ofthe second degree = x\^^ + &- A^) - 2^Bxy + y%A^ + B^ - B^)= B^x^-2fiiBxy + f<Y= (Bx- kyy, a perfect we see that in the equation of any parabola, the termsof the second degree form a perfect square. This is the distinguishing characteristic of the equation of aparabola. 135. Trace the parabola y^ = 8x- 8y, and find (1) the equation ofits axis, (2) the equation of its directrix, (3) the co-ordinates of itsvertex, (4) the co-ordinates of its focus. We shall reduce the given equation to the standard formy^ = iax, by changing the origin. The given equation may be written y^ + &y = 8xor y2 + 83/+16 = 8(« + 2), ^, 124 THE PAEABOLA. [chap. viii. ^ n i^ n ~ — — ~ 7 ■ / / / M / / 0 X / / / / / / / / / / / 1 1 1 / / (O .- 4J X A (- 2_- 4 s X- V \ 1 \ \ \ \ \ \ s V \ S s s s s s \ Changing the origin to the point (- 2, - 4), writing a; - 2for X and y - 4 for y, the equation becomes y^ = 8a;... the new or
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