ENVELOPES 
189 
hypotenuse varies so that the area of the triangle is always the 
same. Find the envelope of the hypotenuse. 
Draw an accurate figure showing a good number of the triangles. 
Take 1 cm. as the unit of length and 2 sq. cm. as the area of the 
triangles. 
4. Circles are drawn on the chords of a parabola which are per 
pendicular to the axis. Show that the envelope is part of an equal 
parabola. 
Make an accurate drawing of a good number of circles for the 
parabola y 2 = x, 1 cm. being taken as the unit of length. 
5. Show that the envelope of all ellipses having coincident axes, 
the distance between two consecutive vertices of any ellipse being 
the same for all the ellipses, is a square. 
6. What is the envelope of all the chords of a circle which are of 
a given length ? 
7. Find the envelope of straight lines drawn perpendicular to 
the normals of a parabola at the points where they cut the axis. 
8. Find the envelope of a circle which is always tangent to the 
axis of x and always has its centre on the parabola y = x 2 . 
9. Show that the envelope of the lines in the second example of 
§ 1 is an arc of a four-cusped hypocycloid. 
10. A straight line moves in such a way that the sum of its inter 
cepts on two rectangular axes is constant. Find its envelope. Draw 
an accurate figure. 
Observe that the equation Va> + Vy = Vi represents a parabola. 
11. Find the envelope of the family of circles which pass through 
the origin and have their centres on the hyperbola xy = 1. 
Aiis. Part of the curve (a; 2 -j- y 2 ) 2 = 16 xy. 
12. The streams of water in a fountain issue from the nozzle, 
which is small, in all directions, but with the same velocity, v 0 . 
Show that, if the nozzle be regarded as a point, the form of the 
fountain is a paraboloid of revolution. 
13. Circles are drawn on those chords of an ellipse as diameter 
which are parallel to an axis of the ellipse. Show that the envelope 
is part of an ellipse, one axis of which is equal to the axis men 
tioned, the other axis being equal to the diagonal of the rectangle 
which circumscribes the ellipse. 
14. Circles are drawn on chords of the hyperbola xy = 1 as diam 
eters. Find their envelope.