What is auxiliary ellipse?

The circle with major axis as diameter is called the auxiliary circle of the ellipse. If a>b, then x2+y2=a2.

Is auxiliary circle concentric with ellipse?

The feet of the perpendiculars drawn from either of the foci to any tangent to the ellipse lies on a circle, concentric with the ellipse is called an auxiliary circle of an ellipse. It is the circle described on the major axis of an ellipse as its diameter.

What is the equation of director circle of an ellipse?

Director circle of an Ellipse The director circle is the locus of the point of intersection of pair of perpendicular tangents to an ellipse. ⇒ x2 + y2 = a2 + b2, which is the equation of the director circle.

What is Auxiliary circle of hyperbola?

The Auxiliary circle of a hyperbola is defined as the circle with the center same as the hyperbola and with transverse axis as it’s diameter. The end points of transverse axis are the two vertices of the hyperbola, so the circle also contains the two vertices of the hyperbola.

What is extremity of ellipse?

The sum of the focal distances of any pint on the ellipse is equal to the major axis. As a result, the distance of focus from the extremity of a minor axis is equal to semi major axis. In parametric form, the equations x = a cos θ and y = b sin θ together represent the ellipse.

How is the equation of an ellipse determined when the foci and vertices are given?

Just as with ellipses centered at the origin, ellipses that are centered at a point (h,k) have vertices, co-vertices, and foci that are related by the equation c2=a2−b2 c 2 = a 2 − b 2 .

How do you write the equation of a circle director?

For any general circle whose equation is ${\left( {x – b} \right)^2} + {\left( {y – c} \right)^2} = {r^2}$, the equation of director circle is given by ${\left( {x – b} \right)^2} + {\left( {y – c} \right)^2} = {\left( {\sqrt 2 r} \right)^2}$.

How do you find the Auxiliary circle of a hyperbola?

The circle described on the transverse axis of a hyperbola as diameter is called its Auxiliary Circle. If x2a2 – y2b2 = 1 is a hyperbola, then its auxiliary circle is x2 + y2 = a2. The transverse axis of the hyperbola x2a2 – y2b2 = 1 is AA’ and its length = 2a.

How do you find the eccentric angle of an ellipse?

Let P be any point on the ellipse. Draw PN perpendicular to the major axis and produce it to meet the auxiliary circle at Q. Then angle ACQ is called the ‘eccentric angle’ of the point P.

What is the auxiliary circle of an ellipse?

Auxiliary circle of an ellipse which is a circle described on the major axis of an ellipse as its diameter. Let the ellipse be x 2 a 2 + y 2 b 2 = 1 …

What is the equation of the ellipse?

Let the equation of the ellipse is. x^2/a^2 + y^2/b^2 =1 , when the major axis of ellipse becomes the diameter of a cicle , it is called auxiliary circle of the ellipse.

How do you find the radius of an auxiliary circle?

The standard equation of an ordinary circle is [math] (x-h)^2+ (y-k)^2=r^2 [/math], where [math]r [/math] is the radius. The center of the auxiliary circle is always the same as the ellipse’s center; the circle itself touches both vertices.

What is the eccentric angle of P on the ellipse?

Through P, draw a line perpendicular to major axis intersecting major axis in N and auxiliary circle in P’. The points P and P’ are called as corresponding points on the ellipse and auxiliary circle respectively. This angle is known as the eccentric angle of the point P on the ellipse and auxiliary circle respectively.