Question 1.
If the point on , where
, where the tangent is parallel to
has an ordinate
, then what is the value of
?
Question 2:
Prove that the segment of the tangent to the curve , which is contained between the coordinate axes is bisected at the point of tangency.
Question 3:
Find all the tangents to the curve for
that are parallel to the line
.
Question 4:
Prove that the curves , where
, and
, where
is a differentiable function have common tangents at common points.
Question 5:
Find the condition that the lines may touch the curve
.
Question 6:
Find the equation of a straight line which is tangent to one point and normal to the point on the curve , and
.
Question 7:
Three normals are drawn from the point to the curve
. Show that c must be greater than 1/2. One normal is always the x-axis. Find c for which the two other normals are perpendicular to each other.
Question 8:
If and
are lengths of the perpendiculars from origin on the tangent and normal to the curve
respectively, prove that
.
Question 9:
Show that the curve , and
is symmetrical about x-axis and has no real points for
. If the tangent at the point t is inclined at an angle
to OX, prove that
. If the tangent at
meets the curve again at Q, prove that the tangents at P and Q are at right angles.
Question 10:
Find the condition that the curves and
intersect orthogonality and hence show that the curves
and
also intersect orthogonally.
More later,
Nalin Pithwa.