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Tangent and Normal Lines
We can define a tangent line to be a line that touches a curve at a point
and has the same slope as the curve at that point. What we are used to seeing is a line that gently rests against the curve so as to pass through the curve only once but keep in mind that a tangent line may pass through the curve as shown at right.
The tangent line at
has equation
A normal line is perpendicular to the tangent line and hence has equation
This is assuming the derivative is not 0 at that point in which case the tangent and normal lines are given by
and respectively.
STEP 2013, Math II, #1: (i) Find the value of
for which the line touches the curve [at a single point.]
If instead the line intersects the curve when
and , where , show that . Show by means of a sketch that .
(ii) The line
, where , intersects the curve when and , where . Show by means of a sketch,
or otherwise, that
.
(iii) Show by means of a sketch that the straight line through the points and , where , intersects the y-axis at a
positive value of
. Which is greater, or ?
(iv) Show, using a sketch or otherwise, that if and
then .
STEP 2000, Math III, #1: Sketch on the same axes the two curves and , given by
:
:
The curves intersect at P and Q. Given that the coordinates of P are (which you need not evaluate), write down the coordinates of Q
in terms of
and .
The tangent to through P meets the tangent to through Q at the point M, and the tangent to through P meets the tangent to through Q at N . Show that the coordinates of M are
and write down the coordinates of N.
Show that PMQN is a square.
STEP 2005, Math I, #2: The point P has coordinates and the point Q has coordinates , where and are non-zero and .
The curve C is given by
. The point R is the intersection of the tangent to C at P and the tangent to C at Q.
Show that R has coordinates
.
The point S is the intersection of the normal to C at P and the normal to C at Q. If
and are such that lies on the line PQ, show
that S has coordinates
, and the quadrilateral PSQR is a rectangle.
STEP 2005, Math III, #5: Let P be the point on the curve (where is nonzero) at which the gradient is
. Show that the equation of the tangent at P is
Show that the curves and (where and are nonzero) have a common tangent with
gradient
if and only if
Show that, in the case , the two curves have exactly one common tangent if and only if they touch each other. In the case ,
find necessary and sufficient conditions for the two curves to have exactly one common tangent.
2008, Math II, #4: A curve is given by
where
is a constant satisfying . Show that the gradient of the curve at the point P with coordinates is
provided , Show that , the acute angle between OP and the normal to the curve at P, satisfies
Show further that, if at P, then:
(i) ;
(ii) ;
(iii) .
Skill Set
Tangent and Normal Lines
No. 1
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