Tangents in a given direction and the angle between curves

What you will be able to do

• find the points where a tangent is horizontal, vertical, parallel or perpendicular to a given line;
• find the tangents to a curve from a point that is not on it;
• find the angle between two curves at a point where they meet;
• decide when two curves touch or cut at a right angle.

Finding where the tangent has a given gradient

To find where the tangent has gradient $k$, solve $f'(x) = k$. Then find $y$ from the curve.

• Horizontal tangent: $f'(x) = 0$.
• Parallel to a line with gradient $m$: $f'(x) = m$.
• Perpendicular to a line with gradient $m$: $f'(x) = -\dfrac{1}{m}$.
• Vertical tangent: $\dfrac{dy}{dx}$ is infinite, so the denominator of $\dfrac{dy}{dx}$ is $0$.

For $y = x^3 - 3x$, the derivative is $3x^2 - 3$. It equals $1$ when $3x^2 = 4$, so $x = \pm\dfrac{2}{\sqrt{3}}$. There are two such points, and the two tangents are parallel to each other.

A tangent from an outside point

A point outside a curve may have two tangents to it. Take the point $(0, -1)$ and the parabola $y = x^2$. We do not know where the tangent touches, so call that point $(t, t^2)$.

1. The tangent at $t$. The gradient is $2t$, so $y - t^2 = 2t(x - t)$, that is, $y = 2tx - t^2$.
2. Make it pass through $(0, -1)$. $-1 = -t^2$, so $t = 1$ or $t = -1$.
3. The tangents. For $t = 1$: $y = 2x - 1$. For $t = -1$: $y = -2x - 1$.

There are two tangents, and they touch the parabola at $(1, 1)$ and $(-1, 1)$. A point inside the parabola has no tangent to it.

The angle between two curves

Two curves that meet at a point make an angle there. It is the angle between their tangents. If the tangents have gradients $m_1$ and $m_2$, the acute angle $\varphi$ between them is given by

Two special cases:

• If $m_1 = m_2$, then $\varphi = 0$. The curves touch.
• If $m_1m_2 = -1$, the tangents are perpendicular. The curves cut at a right angle, and they are called orthogonal.

The curves $y = x^2$ and $y = 2 - x^2$ meet where $x^2 = 1$, so at $(1, 1)$ and $(-1, 1)$. At $(1, 1)$ the gradients are $m_1 = 2$ and $m_2 = -2$. Then

A worked example

Find the points on $y = x^2 - 4x + 5$ where the tangent is perpendicular to the line $x + 2y = 6$.

1. The line's gradient. $x + 2y = 6$ gives $y = -\tfrac{1}{2}x + 3$, so its gradient is $-\tfrac{1}{2}$.
2. The tangent's gradient. It must be $-\dfrac{1}{-1/2} = 2$.
3. Solve $f'(x) = 2$. $2x - 4 = 2$ gives $x = 3$.
4. The point. $y = 9 - 12 + 5 = 2$. The point is $(3, 2)$.

Practise with new numbers

Try it in Python

The program finds the angle between the curves $y = x^2$ and $y = 2 - x^2$ at their meeting point, using the gradients and the formula.

Common mistakes

• Using the gradient of the line as $f(x)$. Set $f'(x)$, not $f(x)$, equal to the gradient.
• Forgetting the point of contact is unknown. For a tangent from an outside point, call the point of contact $(t, f(t))$.
• Taking the obtuse angle. The formula with the absolute value gives the acute angle. The curves also make the angle $180^\circ - \varphi$.
• Dividing by zero in $1 + m_1m_2$. If it is $0$, the angle is $90^\circ$.
• Using the angle formula before finding the point. First find where the curves meet, and then the gradients there.

Remember these

• Tangent parallel to a line: $f'(x) = m$. Perpendicular: $f'(x) = -\dfrac{1}{m}$. Horizontal: $f'(x) = 0$.
• For a tangent from an outside point, let the point of contact be $(t, f(t))$ and solve for $t$.
• $\tan\varphi = \left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|$. Equal gradients mean the curves touch. $m_1m_2 = -1$ means they are orthogonal.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. At which points is the tangent to $y = x^3 - 3x$ horizontal?

2. At which point is the tangent to $y = x^2$ parallel to the line $y = 6x - 1$?

3. At which point is the tangent to $y = x^2$ perpendicular to the line $y = -\tfrac{1}{4}x + 2$?

4. What is the angle between two lines with gradients $1$ and $-1$?

5. The curves $y = x^2$ and $y = x^4$ both pass through the origin. Do they touch or cut there?

References

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 6 (Application of Derivatives).
• OpenStax, Calculus Volume 1, Section 3.1 (Defining the Derivative), the part on tangent lines.