Download PDF by John Casey: A treatise on the analytical geometry of the point, line,
By John Casey
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Additional info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples
A) Determine the equation of the chord joining the points P1 and P2 on the parabola with parameters t1 and t2 , respectively, where t1 and t2 are unequal and non-zero. y Q P1 –a F (a, 0) x Now assume that the chord P1 P2 passes through the focus (a, 0) of the parabola. P2 directrix (b) Prove that t1 t2 = −1. (c) Use the result of Example 2(a) to write down the equations of the tangents to the parabola at P1 and P2 , and to prove that these tangents are perpendicular. (d) Find the point of intersection P of the two tangents in part (c), and verify that it lies on the directrix x = −a of the parabola.
Determine the centre of those that have a centre. (a) 11x 2 + 4xy + 14y 2 − 4x − 28y − 16 = 0 (b) x 2 − 4xy + 4y 2 − 6x − 8y + 5 = 0 In fact, using the above strategy we can prove the following result. Theorem 3 A non-degenerate conic with equation We omit a proof of this result. Ax2 + Bxy + Cy2 + Fx + Gy + H = 0 and matrix A = A 1 2B 1 2B C Since det A = AC − 1 B 2 = − 1 (B 2 − 4AC). 4 4 Theorem 3 is often referred to as ‘the B 2 − 4AC test’ for conics. can be classified as follows: (a) If det A < 0, E is a hyperbola.
A F′ (–ae, 0) T F (ae, 0) x It follows that TF (a/ cos t) − ae 1 − e cos t = = . TF (a/ cos t) + ae 1 + e cos t We deduce that TF PF PF PF = , or . = PF TF TF TF By applying the Sine Formula to the triangles PFT and PF T , we obtain that PF sin ∠PTF sin ∠PTF PF = , = and TF sin ∠T PF TF sin ∠TPF so that sin ∠PTF sin ∠PTF . = sin ∠TPF sin ∠TPF Since ∠PTF = ∠PTF it follows that sin ∠TPF = sin ∠TPF , so that ∠TPF = π − ∠TPF since ∠TPF = ∠TPF . Hence ∠TPF equals the angle denoted by the symbol α in the diagram, and this is equal to the angle β (as α and β are vertically opposite).
A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples by John Casey