Class 10 Maths · Chapter 10 · Topic 1 of 4
A tangent touches once; a secant crosses twice
Use perpendicular distance to distinguish tangent from secant.
Ideas you’ll use
- Tangent
- A line with exactly one common point with a circle. That point is the point of contact. A secant has two common points.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
Move the horizontal line: zero, one or two contacts
y-coordinate (unit)
x-coordinate (unit)
Equal units on both axes.
- Circle · Original coordinate construction.
- No intersection · Original coordinate construction.
- Tangent · Original coordinate construction.
- Secant · Original coordinate construction.
- 1. One contact · (0, 5)
- 2. Secant endpoint · (5, 0)
Assumptions: Both axes use the same length per unit. Straight segments join the stated coordinates. The algebra gives exact lengths.
Read the plotted values
Circle
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| 5 | 0 |
| 4.957224306869052 | 0.6526309611002579 |
| 4.8296291314453415 | 1.2940952255126037 |
| 4.619397662556434 | 1.913417161825449 |
| 4.330127018922194 | 2.4999999999999996 |
| 3.9667667014561756 | 3.043807145043603 |
| 3.5355339059327378 | 3.5355339059327373 |
| 3.043807145043603 | 3.9667667014561756 |
| 2.5000000000000004 | 4.330127018922193 |
| 1.9134171618254492 | 4.619397662556434 |
| 1.2940952255126037 | 4.8296291314453415 |
| 0.6526309611002585 | 4.957224306869052 |
| 3.061616997868383e-16 | 5 |
| -0.652630961100258 | 4.957224306869052 |
| -1.2940952255126033 | 4.8296291314453415 |
| -1.9134171618254476 | 4.619397662556434 |
| -2.499999999999999 | 4.330127018922194 |
| -3.043807145043603 | 3.9667667014561756 |
| -3.5355339059327373 | 3.5355339059327378 |
| -3.966766701456175 | 3.0438071450436044 |
| -4.330127018922194 | 2.4999999999999996 |
| -4.619397662556434 | 1.9134171618254494 |
| -4.829629131445341 | 1.294095225512605 |
| -4.957224306869052 | 0.65263096110026 |
| -5 | 6.123233995736766e-16 |
| -4.957224306869052 | -0.6526309611002589 |
| -4.8296291314453415 | -1.294095225512604 |
| -4.619397662556434 | -1.9134171618254483 |
| -4.3301270189221945 | -2.499999999999999 |
| -3.9667667014561756 | -3.043807145043603 |
| -3.5355339059327395 | -3.5355339059327355 |
| -3.0438071450436044 | -3.9667667014561747 |
| -2.500000000000002 | -4.330127018922192 |
| -1.9134171618254476 | -4.619397662556434 |
| -1.2940952255126033 | -4.8296291314453415 |
| -0.6526309611002581 | -4.957224306869052 |
| -9.184850993605148e-16 | -5 |
| 0.6526309611002563 | -4.957224306869053 |
| 1.2940952255126015 | -4.829629131445342 |
| 1.9134171618254459 | -4.619397662556435 |
| 2.5000000000000004 | -4.330127018922193 |
| 3.0438071450435995 | -3.9667667014561787 |
| 3.535533905932737 | -3.5355339059327386 |
| 3.9667667014561747 | -3.0438071450436044 |
| 4.330127018922192 | -2.500000000000002 |
| 4.619397662556434 | -1.9134171618254479 |
| 4.829629131445341 | -1.294095225512608 |
| 4.957224306869052 | -0.6526309611002584 |
| 5 | -1.2246467991473533e-15 |
No intersection
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| -7 | 7 |
| 7 | 7 |
Tangent
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| -7 | 5 |
| 7 | 5 |
Secant
| x-coordinate (unit) | y-coordinate (unit) |
|---|---|
| -7 | 0 |
| 7 | 0 |
Radius 5: the line y=7 misses; y=5 touches; y=0 crosses twice.
Compare a line nearer than the radius
If its perpendicular distance to the centre is less than r, it cuts twice.
distance<r ⇒ secantIn the picture: Locate both intersection points.
Read all the reasoning together
Compare a line nearer than the radius
If its perpendicular distance to the centre is less than r, it cuts twice.
distance<r ⇒ secantMove to exactly one radius
At distance r the line has one contact point.
distance=r ⇒ tangentMove farther away
Beyond r there is no intersection.
distance>r ⇒ outside lineKeep tangent counts clear
There is one tangent at a fixed contact point, but infinitely many contact points around the circle.
One tangent per point on the circle
Use perpendicular distance to distinguish tangent from secant.
Worked example: Three positions of one direction
Distance from centre to line: less than r ⇒ secant; equal to r ⇒ tangent; greater than r ⇒ no intersectionThe distance here is the perpendicular distance. Two tangents parallel to a given direction lie on opposite sides of the circle.
Read the supporting explanation
A line outside the circle has no common point. If we move a secant parallel to itself towards the edge, its two intersection points get closer until they coincide at contact. A circle has infinitely many tangents overall, but only one at a chosen point.
Go deeper: Tangent versus tangent segment
A tangent line continues forever. Its “length from P” means the finite segment from an external point P to its point of contact, not the length of the whole line.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Can a circle have three parallel tangents?
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