## Download PDF by John Casey: A treatise on the analytical geometry of the point, line,

By John Casey

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**Read or Download 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 PDF**

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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**

**Sample text**

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

by John

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