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What is the Condition for a Line to be Tangent to a Circle?
Grade Level:
Class 10
AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine
Definition
What is it?
A line is tangent to a circle if it touches the circle at exactly one point. The main condition for this to happen is that the perpendicular distance from the center of the circle to the line must be equal to the radius of the circle.
Simple Example
Quick Example
Imagine a car tyre (circle) resting perfectly on a flat road (line). The road touches the tyre at only one point. The distance from the center of the tyre to the road is exactly the radius of the tyre. If the road was further away, it wouldn't touch; if it cut through, it would be a different kind of line.
Worked Example
Step-by-Step
PROBLEM: A circle has its center at (0,0) and a radius of 5 units. Is the line 3x + 4y = 25 tangent to this circle?
STEP 1: Identify the center (h,k) and radius (r) of the circle. Here, (h,k) = (0,0) and r = 5.
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STEP 2: Identify the equation of the line in the form Ax + By + C = 0. Here, 3x + 4y - 25 = 0. So, A=3, B=4, C=-25.
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STEP 3: Calculate the perpendicular distance (d) from the center (h,k) to the line using the formula: d = |Ah + Bk + C| / sqrt(A^2 + B^2).
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STEP 4: Substitute the values: d = |(3)(0) + (4)(0) - 25| / sqrt(3^2 + 4^2).
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STEP 5: Calculate the value: d = |-25| / sqrt(9 + 16) = 25 / sqrt(25) = 25 / 5 = 5.
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STEP 6: Compare the distance (d) with the radius (r). Here, d = 5 and r = 5. Since d = r, the line is tangent to the circle.
ANSWER: Yes, the line 3x + 4y = 25 is tangent to the circle.
Why It Matters
Understanding tangents is crucial in engineering for designing gears and wheels, and in physics for studying projectile motion. It's also used in AI/ML for optimizing paths, and in space technology for calculating satellite orbits to avoid collisions. Engineers and scientists use this concept daily.
Common Mistakes
MISTAKE: Forgetting the absolute value in the distance formula, leading to negative distances. | CORRECTION: Always use the absolute value | | around the numerator |Ah + Bk + C| because distance cannot be negative.
MISTAKE: Incorrectly identifying A, B, and C from the line equation, especially if not in Ax + By + C = 0 form. | CORRECTION: First rearrange the line equation to Ax + By + C = 0 before picking out A, B, and C. For example, x - 2y = 5 should be x - 2y - 5 = 0.
MISTAKE: Confusing the radius with the diameter, or squaring the radius instead of using the radius directly in comparison. | CORRECTION: Always ensure you are using the actual radius (r) of the circle for comparison with the calculated perpendicular distance (d).
Practice Questions
Try It Yourself
QUESTION: A circle has center (1,2) and radius 3. Is the line x - y + 4 = 0 tangent to this circle? | ANSWER: Yes
QUESTION: Find the value of 'k' if the line 4x + 3y + k = 0 is tangent to the circle x^2 + y^2 = 25. | ANSWER: k = 25 or k = -25
QUESTION: A circle passes through the origin (0,0) and has its center at (3,4). Find the equation of the tangent to this circle at the point (0,0). | ANSWER: 3x + 4y = 0
MCQ
Quick Quiz
What is the condition for a line to be tangent to a circle?
The distance from the center to the line is greater than the radius.
The distance from the center to the line is less than the radius.
The distance from the center to the line is equal to the radius.
The line passes through the center of the circle.
The Correct Answer Is:
C
For a line to be tangent, it must touch the circle at exactly one point. This happens precisely when the perpendicular distance from the circle's center to the line is exactly equal to the radius.
Real World Connection
In the Real World
When a drone delivers a package, its path (line) might need to avoid certain no-fly zones (circles). Tangent calculations help plan the most efficient path that just skims the edge of the zone without entering it, ensuring safe and quick delivery by companies like Swiggy or Zomato.
Key Vocabulary
Key Terms
TANGENT: A line that touches a circle at exactly one point | RADIUS: The distance from the center of a circle to any point on its circumference | PERPENDICULAR DISTANCE: The shortest distance from a point to a line | EQUATION OF A LINE: A mathematical expression defining all points on a straight line | CENTER OF A CIRCLE: The midpoint of a circle, equidistant from all points on its circumference
What's Next
What to Learn Next
Great job learning about tangents! Next, you can explore 'Properties of Tangents to a Circle' to understand more special rules and theorems that apply. This will build on what you've learned and open up new ways to solve geometry problems.
