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# Prove some conditions by which a conic section is an ellipse or a parabola

Consider a conic section with eccentricity , a focus at the origin, and with a vertical directrix which is a distance to the left of .

1. Prove that if is an ellipse or parabola then every point of lies to the right of and satisfies the equation (in polar coordinates) 2. Prove that if the conic section is a hyperbola, point on the right branch satisfy the equation in part (a) while points on the left branch satisfy 1. Proof. If is a conic section with eccentricity and focus at the origin, then we know the points on are given by those points such that where is the directrix at a distance to the left of . So, if has polar coordinates and , we have and . Therefore, if lies to the right of the directrix (so since in this case), and since , we must have 2. Proof. If is a hyperbola, then we know and points on the left branch still satisfy just as in part (a). For the points on to the right of , we have so implies 1. • 