How to Find the Equation of a Tangent to a Circle?
The concept of tangent of a circle plays a key role in geometry and is widely applicable in both exams (like CBSE, ICSE, JEE) and real-world situations, such as engineering, design, and daily problem-solving. Understanding tangents helps students quickly solve circle geometry questions and develop strong logical reasoning skills.
What Is Tangent of a Circle?
A tangent of a circle is a straight line that touches a circle at exactly one point, called the point of contact. Unlike a secant, which cuts the circle at two points, or a chord, which lies inside the circle, a tangent only 'touches' the circle from the outside and never enters its interior. You’ll find this concept applied in areas such as circle geometry, coordinate geometry, and real-world architectural designs.
Key Formula for Tangent of a Circle
Here’s the standard formula for the equation of a tangent to the circle \( x^2 + y^2 = r^2 \) at point \((x_1, y_1)\):
\( x x_1 + y y_1 = r^2 \)
To find the length of a tangent from an external point \( P(x_0, y_0) \) to the circle, use:
Length of tangent = \( \sqrt{(x_0 - h)^2 + (y_0 - k)^2 - r^2} \), where (h, k) is the center.
Key Properties of a Tangent of a Circle
- The tangent touches the circle at only one point (point of contact).
- It is always perpendicular to the radius at the point of tangency.
- From a single point outside the circle, exactly two tangents can be drawn.
- The lengths of tangents from an external point to a circle are equal.
Tangent Theorems Explained
- Tangent-Radius Theorem: At the point of tangency, the radius is perpendicular to the tangent.
If O is the center and P is the point of tangency, then \( OP \perp \text{tangent at } P \). - Two-Tangents Theorem: Tangents drawn from an external point are equal in length.
Equation of Tangent by Slope and Point
For a circle \( (x-h)^2 + (y-k)^2 = r^2 \):
- At point \( (x_1, y_1) \): \( (x_1-h)(x-h) + (y_1-k)(y-k) = r^2 \)
- With slope \( m \): \( y = mx \pm r\sqrt{1 + m^2} \) (for center at origin)
Construction: How to Draw a Tangent to a Circle
- Draw the circle with center O and mark the point P where tangent is to be drawn.
- Join OP (radius).
- At point P, draw a straight line perpendicular to OP using a compass or ruler—this is the tangent.
This method is simple and is needed in geometry projects and exams. For extra visuals, refer to Vedantu’s geometry classes for step-by-step compass construction.
Step-by-Step Illustration
1. Given a circle with equation \( x^2 + y^2 = 25 \) and point of contact (3, 4).2. Equation of tangent: Substitute into formula for the circle at the point \( (x_1, y_1) \):
3. \( 3x + 4y = 25 \) is the equation of tangent at (3,4) to the circle.
Speed Trick or Vedic Shortcut
If you want to quickly check if a line is a tangent to a circle in coordinate geometry, substitute the coordinates of the proposed tangency point into the line and circle equations. If both are satisfied, it’s correct!
Example Trick: For circle with center (h,k) and radius r, and a line \( y = mx + c \):
- Put distance from (h,k) to line = r.
\( \frac{|mh - k + c|}{\sqrt{1 + m^2}} = r \)
This shortcut saves time during board exams and competitive tests. Vedantu includes more tips like these in live sessions to help students master geometry quickly!
Try These Yourself
- Find the equation of the tangent to the circle \( x^2 + y^2 = 36 \) at the point (6,0).
- From point (8,6), draw two tangents to the circle \( x^2 + y^2 = 16 \). What is their length?
- Show that the line \( 4x + 3y = 24 \) is tangent to the circle \( x^2 + y^2 = 36 \).
- Construct a tangent to the circle with center (0,0) and radius 5 cm at point (3,4).
Frequent Errors and Misunderstandings
- Mixing up the tangent and the chord or secant.
- Forgetting that tangent is always perpendicular to the radius at the point of contact.
- Using the wrong equation for circles not centered at the origin.
- Not setting the correct length formula for tangents from external points.
- Plotting the tangent at the wrong point of contact.
Relation to Other Concepts
The tangent of a circle connects with chords of a circle, secants, radius, and angle properties. It is also key when studying circle theorems and understanding the difference between various straight lines related to the circle.
Classroom Tip
A quick way to remember a tangent is always a 'just touch' line and not a 'cut' line. To remember perpendicularity: Picture the radius as an arm, and the tangent as a tight slap—always at 90°! Vedantu's teachers often use mnemonics and visuals for such tricky geometry facts.
We explored tangent of a circle—from definition, important formulas, examples, common errors, and its link to other concepts. Practice with more questions and Vedantu’s worksheets to gain confidence and ace your exams!
Further Learning: Internal Links
- Equation of a Circle – Learn to derive tangent and normal equations.
- Chord of Circle – Distinguish tangent from chord with easy graphs.
- Secant of a Circle – Get a clear difference between tangent and secant lines.
- Properties of Circle – Master all properties including tangency, chord, and radius relations.
FAQs on Tangent of a Circle: Meaning, Theorem, Construction & Formula
1. What is a tangent to a circle?
A tangent to a circle is a straight line that touches the circle at exactly one point, called the point of tangency. This point is the only intersection between the line and the circle. The tangent line never enters the interior of the circle.
2. What is the relationship between a tangent and the radius at the point of contact?
The radius drawn to the point of tangency is always perpendicular to the tangent line. This is a fundamental property used in many circle geometry problems.
3. How do you find the length of a tangent from an external point to a circle?
If two tangents are drawn to a circle from the same external point, their lengths are equal. To find the length, consider the right-angled triangle formed by the external point, the center of the circle, and the point of tangency. You can use the Pythagorean theorem to calculate the tangent length.
4. What is the equation of a tangent to a circle?
The equation of a tangent to a circle depends on the given information. If you know the point of tangency (x₁, y₁) and the radius (r), the equation is: x₁x + y₁y = r². Other methods exist to derive the equation if the point of tangency and circle equation or slope and a point are known.
5. How do you construct a tangent to a circle from an external point?
1. Draw a circle with center O and radius r. 2. Mark an external point P. 3. Draw a line segment OP. 4. Find the midpoint M of OP. 5. Draw a circle with center M and radius OM. 6. This circle intersects the original circle at points A and B. 7. Draw lines PA and PB. These are the tangents to the circle from P.
6. What are the main theorems related to tangents of a circle?
The two most important theorems are: 1. The Tangent-Radius Theorem: A tangent to a circle is perpendicular to the radius at the point of contact. 2. The Two Tangents Theorem: The lengths of the two tangents drawn from an external point to a circle are equal.
7. What is the difference between a tangent, a secant, and a chord?
• A chord connects two points on the circle. • A secant is a line that intersects a circle at two points. • A tangent touches a circle at only one point.
8. How are tangents used in calculus?
In calculus, the concept of a tangent is fundamental to understanding derivatives. The derivative of a function at a point represents the slope of the tangent line to the function's graph at that point.
9. Can you give an example of a real-world application of tangents?
Tangents are used in many engineering and design applications. For example, the design of gears and cams relies on understanding tangents to ensure smooth and efficient operation.
10. How can I solve problems involving tangents and circles?
Solving problems often involves applying the tangent-radius theorem, the two tangents theorem, and using geometric properties of triangles and circles. Drawing accurate diagrams is crucial.
11. What are some common mistakes students make when dealing with tangents?
Common mistakes include confusing tangents with chords or secants, forgetting that the radius is perpendicular to the tangent at the point of contact, and incorrectly applying the Pythagorean theorem.
12. How many tangents can be drawn to a circle from a point outside the circle?
Two tangents can be drawn to a circle from a point outside the circle.
