Blaga P. Lectures on the differential geometry of - tiera.ru

26. Show that all the normals to the curve
⎧
⎪⎨ x = a(cos t + sin t)
⎪⎩ y = a(sin t − cos t)
lie at the same distance with respect to the origin.
27. Show that the segment from the normal to the curve
⎧
⎪⎨ x = 2a sin t + a sin t cos
⎪⎩
2 t
y = −a cos3 t
Problems 107
intercepted between the coordinate axes has constant length, equal to 2a.
28. It is possible to draw two pairs of tangents to the Bernoulli’s lemniscate (x 2 + y 2 ) 2 =
a 2 (x 2 − y 2 ), parallel to a given direction. Show that the straight lines connecting the
tangency points of each pair intercept an angle of 120 ◦ .
29. Prove that the circles
intersect each other under a right angle iff
x 2 + y 2 + a1x + b1y + c1 = 0
x 2 + y 2 + a2x + b2y + c2 = 0
a1a2 + b1b2 = 2(c1 + c2).
30. Show that if all the normals to a given curve are passing through the same point,
then the curve is an arc of circle.
31. Find the equation of the curve from whose points a given parabola is seen under a
constant angle, α. Which curve do we get for α = π
2 ?.
32. Find the envelope of a family of straight lines passing through the extremities of a
pair of conjugates diameters of an ellipse.
33. Find the envelope of a family of straight lines which determine on a right angle a a
triangle of constant area.
34. Find the envelope of a family of straight lines which determine on a right angle a a
triangle of constant perimeter.