What Is The Tangent Of A Circle Formula Properties And Solved Examples
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 Explained With Definition And Formula
1. What is a tangent of a circle?
A tangent of a circle is a straight line that touches the circle at exactly one point called the point of contact.
- It intersects the circle at only one point.
- The radius drawn to the point of contact is perpendicular to the tangent.
- It does not cut through the circle like a secant.
2. What is the condition for a line to be tangent to a circle?
A line is tangent to a circle if the distance from the center to the line equals the radius.
- For a circle with center (h, k) and radius r.
- Find the perpendicular distance from (h, k) to the line Ax + By + C = 0.
- If distance = r, the line is a tangent.
3. What is the formula of the tangent to a circle?
The equation of the tangent to the circle x² + y² = r² at point (x₁, y₁) is xx₁ + yy₁ = r².
- The point (x₁, y₁) must lie on the circle.
- This formula is derived using the point-slope concept.
- It applies specifically to circles centered at the origin.
4. How do you find the equation of a tangent to a circle at a given point?
To find the equation of a tangent at a given point, use the standard tangent formula for that circle.
- Step 1: Verify the point lies on the circle.
- Step 2: For x² + y² = r², use xx₁ + yy₁ = r².
- Step 3: Substitute the given point (x₁, y₁).
5. Why is the radius perpendicular to the tangent?
The radius is perpendicular to the tangent at the point of contact because the shortest distance from the center to the tangent line is the radius.
- The tangent touches the circle at one point only.
- The perpendicular from the center to the tangent meets it at the point of contact.
- This forms a 90° angle.
6. How many tangents can be drawn from a point to a circle?
From an external point, exactly two tangents can be drawn to a circle.
- From a point on the circle: only one tangent.
- From a point inside the circle: no tangents.
7. What is the length of the tangent from an external point?
The length of the tangent from an external point is given by √(OP² − r²), where OP is the distance from the center to the point.
- O = center of circle
- P = external point
- r = radius
8. What is the difference between a tangent and a secant?
A tangent touches the circle at one point, while a secant cuts the circle at two points.
- Tangent → one point of contact.
- Secant → enters and exits the circle.
- A tangent is a limiting position of a secant.
9. Can you give an example of finding a tangent to a circle?
Yes, for the circle x² + y² = 16, the tangent at (4, 0) is 4x = 16 or x = 4.
- Step 1: Identify r² = 16.
- Step 2: Use formula xx₁ + yy₁ = r².
- Step 3: Substitute (4, 0) → 4x + 0 = 16.
10. What are the key properties of tangents to a circle?
The key properties of tangents to a circle describe their geometric behavior and equal lengths.
- The tangent is perpendicular to the radius at the point of contact.
- Tangents from the same external point are equal in length.
- A circle can have infinitely many tangents, but only one at each point.
