One Point, More Than One Visit
A parametric curve may pass through the same point at different parameter values. The point might be a crossing, where two branches meet, or the parametrization might trace the same geometric path more than once. Either way, a single point on the plane can correspond to several parameter values.
At each visit, the curve has its own local direction of travel. To understand the geometry at the shared point, calculate the tangent at every parameter value that reaches it. The resulting tangent lines may be different, or several visits may produce the very same line.
Finding the Parameter Values and Tangent Slopes
Suppose a curve is given by \(x=x(t)\) and \(y=y(t)\). To find parameter values that reach a particular point \((a,b)\), solve the two equations \(x(t)=a\) and \(y(t)=b\) together, keeping only parameter values in the stated domain. To check whether the curve reaches the same point at multiple parameter values, compare the coordinate pairs produced by distinct values of \(t\).
At a parameter value \(t_0\) where \(x'(t_0)\ne 0\), the parametric slope formula is
Use the formula separately at each parameter value that reaches the shared point. If the resulting slope is \(m\), the tangent line at \((a,b)\) has equation \(y-b=m(x-a)\). When two visits reach the same point, equal slopes give the same tangent line; unequal slopes give distinct tangent lines.
The condition \(x'(t_0)\ne 0\) matters: the quotient formula cannot be used at a parameter value where its denominator is zero. The examples below use parameter values where the formula applies.
Why the Parametric Slope Formula Works
The formula compares the change in \(y\) with the change in \(x\) as the parameter changes. Here is a direct derivation at a parameter value \(t_0\), assuming \(x'(t_0)\ne 0\). For nearby \(t\ne t_0\), factor the difference quotient as follows:
As \(t\) approaches \(t_0\), the numerator on the right approaches \(y'(t_0)\), and the denominator approaches \(x'(t_0)\). Since \(x'(t_0)\ne 0\), the denominator's limit is nonzero, so the quotient approaches \(y'(t_0)/x'(t_0)\). This is the slope of the curve at that visit. Repeating the calculation at another parameter value reaching the same point gives the tangent direction for that other visit.
A Reliable Comparison Process
Find the distinct parameter values in the domain that give the shared coordinate pair.
Calculate \(x'(t)\) and \(y'(t)\), then evaluate them at each visit.
Where \(x'(t)\ne 0\), divide \(y'(t)\) by \(x'(t)\) at that parameter value.
Use the shared point in point-slope form. Count distinct equations, not parameter values.
Worked Example: A Crossing with Two Different Tangents
Worked Example: A Crossing with Two Different Tangents
Consider \(x(t)=t^2-1\) and \(y(t)=t^3-t\). Find the tangent lines at the point reached when \(t=-1\) and \(t=1\).
Step 1: Check the shared point. At \(t=-1\),
At \(t=1\),
Both parameter values reach \((0,0)\).
Step 2: Differentiate.
Step 3: Find the slope at each visit. At \(t=-1\),
At \(t=1\),
Both denominators are nonzero, so both slope calculations are valid.
Step 4: Write and check the tangent lines. Through \((0,0)\), the lines are
The point \((0,0)\) satisfies both equations. Their slopes, \(-1\) and \(1\), differ, so the curve has two distinct tangent lines at this self-intersecting point.
Worked Example: Two Visits but One Tangent Line
Worked Example: Two Visits but One Tangent Line
Consider \(x(t)=t^2-1\) and \(y(t)=(t^2-1)^2\). Compare the tangents at the point reached when \(t=-1\) and \(t=1\).
Step 1: Check the point. At either \(t=-1\) or \(t=1\), \(t^2-1=0\), so
Both visits are at \((0,0)\).
Step 2: Differentiate. By the chain rule,
Step 3: Evaluate the slopes. At \(t=-1\),
At \(t=1\),
Step 4: Compare the tangent lines. At \((0,0)\), slope zero gives \(y=0\) at both visits. Substituting \((0,0)\) into this equation confirms that it passes through the point. Thus there are two parameter values but only one distinct tangent line.
Worked Example: Three Visits and Three Tangents
Worked Example: Three Visits and Three Tangents
Let \(x(t)=t^3-3t\) and \(y(t)=t^2(t^2-3)\). Find the tangent lines at the origin and determine how many are distinct.
Step 1: Find every parameter value reaching the origin. The equation \(x(t)=0\) factors as
so \(t=0\), \(t=\sqrt{3}\), or \(t=-\sqrt{3}\). At each of these values, \(y(t)=t^2(t^2-3)=0\). Conversely, any visit to the origin must satisfy \(x(t)=0\), so these are exactly the three parameter values.
Step 2: Differentiate.
Step 3: Calculate each slope. At \(t=0\),
At \(t=\sqrt{3}\),
At \(t=-\sqrt{3}\),
Step 4: Write and check the lines. Since all three tangents pass through \((0,0)\), their equations are
Each equation contains the origin, and the three slopes are distinct. There are therefore three distinct tangent lines at the origin.
A Common Mistake: Counting Visits Instead of Lines
A frequent error is to assume that two parameter values necessarily mean two tangent lines. For the curve in the second example, someone might reason: “The origin occurs at \(t=-1\) and \(t=1\), so there must be two tangents.” That conclusion does not follow. The slope calculations give \(0\) at both visits, and both tangent equations are \(y=0\). The correct count is one distinct line.
The reverse check is just as important: do not stop after finding one slope at the shared point. In the first example, calculating only the slope at \(t=1\) would give \(y=x\) and miss the other tangent \(y=-x\). A reliable answer lists every relevant parameter value, its slope, and the resulting line.
Check Your Understanding
For each question, identify the parameter values at the shared point before comparing tangent slopes.
- For \(x(t)=t^2-4\) and \(y(t)=t^3-4t\), which parameter values reach \((0,0)\), and what tangent line occurs at each?
- If two visits to one point have the same slope and both slope calculations are valid, how many distinct tangent lines do they determine?
- For \(x(t)=t^2\) and \(y(t)=t^3\), compare the slopes at the origin reached at \(t=0\) and \(t=0\). What limitation should you notice about using the quotient formula there?
- Why must parameter values outside a stated parameter interval be excluded, even if they produce the same coordinates?
- A point is reached at three parameter values, and the calculated slopes are \(2\), \(2\), and \(-1\). How many distinct tangent lines result?
