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Circle and Line in a Plane

In a parabola...

Question

In a parabola show that the tangent at any point makes equal angles with the focal radius of the point and the line parallel to the axis through the point.

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Q. If a tangent at a point P on a parabola makes angle a with its focal distance, then angle between the tangent and axis of the parabola is:

Q. If the tangent at any point of an ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

α

with the major axis and an angle

β

with the focal radius of the point of contact then show that the eccentricity 'e' of the ellipse is given by the absolute value of

\frac{cos β}{cos α}

Q. If T be any point on the tangent at any point P of a parabola, and if TL be perpendicular to the focal radius SP and TN be perpendicular to the directrix, prove that SL = TN.

Q. If the normals drawn from any point to the parabola cut the line

x = 2 a

in points whose ordinates are in arithmetical progression, prove that the tangents of the angles which the normals make with the axis are in geometrical progression.

Q. The point of contact of the tangent to the parabola

y^{2} = 9 x

which passes through the point

(4, 10)

and makes an angle

θ

with the axis of the parabola such that

tan θ > 2

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