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Equation of Tangent and Normal Lines: Circle: Implicit Differentiation
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Find the equations of the tangent line and normal line to the graph of
.
How do we find the slope of the tangent line?
We need the derivative,
How can we get that?
One way is to use implicit differentiation.
Which rule of differentiation can we use here?
The Power Rule.
Do the implicit differentiation.
Solve for
We get
Is this the slope of the tangent line at
?
No.
How do we get the slope at this point?
We need to evaluate
.
Do that.
Does this solve the problem?
No. This is the slope,
of the tangent line. To get the equation of the tangent line we need its
y
-intercept,
, so we can state the equation of the line as
How do we find
?
Since the line passes through the point of tangency,
, we can substitute those coordinates and our value of
in
and solve for
.
Do the substitution.
becomes
Solve for
.
We get
.
Combine these results to get the equation of the tangent line.
Using these values of
and
, the equation of the
tangent line
at the point (4,3) is
How do we get the equation of the normal line?
We need to get its slope and use the point(4,3) as above.
How are the normal line and the tangent line related geometrically?
They are mutually perpendicular.
How are the slopes of perpendicular lines related?
They are negative reciprocals of each other.
Apply this fact to express the slope of the normal line,
, in terms of
.
Substitute to determine
.
Set up the point-slope equation to find the intercept of the normal line,
.
Use these results to state the equation of the normal line.
The equation of the
normal line
is
.
Let's do a graphical check. What is the shape of the graph of
It is the equation of a circle.
Graph this on paper. Then check your graph by clicking on "Next".
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Add the point
. Then check your graph by clicking on "Next".
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Add the tangent line to your graph. Then check by clicking "Next".
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Does the slope appear to be correct?
Yes, it is negative and about equal to
Does the
y
-intercept appear to be correct?
Yes, the intercept is close to the expected 8.3.
Let's also check the normal line. Add it to your graph.
alt="Diagram of a circle with tangent and normal lines. Your browser understands the <APPLET> tag but isn't running the applet, for some reason." Your browser is completely ignoring the <APPLET> tag!
Does the slope appear to be correct?
Yes, the rise over run for the normal line is 3 over 4, in agreement.
Does the
y
-intercept appear to be correct?
Yes, all of the normal lines of a circle pass through the center of the circle.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
General Contents
Detailed Contents
Index