Analytic Geometry — Equation of a Tangent to a Circle

Analytic Geometry — Equation of a Tangent to a Circle. Practice questions to deepen understanding of the equation of a tangent to a circle. Online math practice with full solutions and step-by-step explanations.

Equation of a tangent to a circle practice — tangent at a point on the circle, tangent from an external point, slopes of the radius and the tangent. Analytic geometry.

10 questions

Question 1
10.00 pts

📏 Tangent to a circle:

What is the condition for a line to be tangent to a circle?

Explanation:
💡 Detailed Explanation:

Tangent — two equivalent conditions! 📏

📐 Geometric condition:
The perpendicular distance from the centre to the line equals the radius:
d(centre, line) = r

📐 Algebraic condition:
Substituting the line into the circle gives a quadratic with a repeated root:
Δ = 0

🎯 Distance formula:
For a line Ax + By + C = 0 and centre (x₀, y₀):
d = |A·x₀ + B·y₀ + C| / √(A² + B²)

📊 Three line positions:
• d < r → secant (2 intersections, Δ > 0)
• d = r → tangent (1 intersection, Δ = 0)
• d > r → no intersection (Δ < 0)

💡 Tangent direction: always perpendicular to the radius at the point of tangency.
Question 2
10.00 pts

➡️ Horizontal tangent:

Find the horizontal tangent lines to the circle \(x^2+(y-3)^2=16\).

Explanation:
💡 Detailed Explanation:

Horizontal tangents from centre + radius! ➡️

📐 Identify centre and radius:
x² + (y − 3)² = 16 → centre (0, 3), radius 4.

🎯 Horizontal tangents:
y = b ± r = 3 ± 4 → y = 7 and y = −1.

📊 Verification:
For y = 7: distance |7 − 3| = 4 = r ✓
For y = −1: distance |−1 − 3| = 4 = r ✓

💡 General rule: for circle (x − a)² + (y − b)² = r²:
• Horizontal tangents: y = b ± r
• Vertical tangents: x = a ± r
Two of each, always.
Question 3
10.00 pts

📏 Vertical tangent:

Find the vertical tangent lines to the circle \((x-2)^2+y^2=9\).

Explanation:
💡 Detailed Explanation:

Vertical tangents from centre + radius! 📏

📐 Centre and radius:
(x − 2)² + y² = 9 → centre (2, 0), radius 3.

🎯 Vertical tangents:
x = a ± r = 2 ± 3 → x = 5 and x = −1.

💡 Memory aid: "Horizontal tangents: y = b ± r" and "Vertical tangents: x = a ± r". Always two of each.
Question 4
10.00 pts

📍 At a given point:

Find the equation of the tangent to the circle \(x^2+y^2=25\) at the point \((3,4)\).

Explanation:
💡 Detailed Explanation:

Tangent at a known point on a circle! 📍

📐 The "split formula":
For a circle x² + y² = r² and a point (x₀, y₀) on the circle, the tangent line is:
x₀·x + y₀·y = r²

Here: 3x + 4y = 25.

🎯 Why this formula works:
The radius from the origin to (3, 4) has direction (3, 4). The tangent at that point is perpendicular to the radius, so its equation is 3·(x − 3) + 4·(y − 4) = 0, simplifying to 3x + 4y = 25.

✅ Verify:
(3, 4) on circle: 9 + 16 = 25 ✓
(3, 4) on line: 9 + 16 = 25 ✓

💡 General version (shifted circle):
For (x − a)² + (y − b)² = r² and tangency point (x₀, y₀):
(x₀ − a)(x − a) + (y₀ − b)(y − b) = r²
Question 5
10.00 pts

📐 Slope:

Find the tangent lines to the circle \(x^2+y^2=5\) with slope \(m=2\).

Explanation:
💡 Detailed Explanation:

Tangent lines with a prescribed slope! 📐

📐 Approach (Δ = 0 method):
The general line with slope 2: y = 2x + n. Substitute into x² + y² = 5:
x² + (2x + n)² = 5
5x² + 4nx + (n² − 5) = 0

For tangency, Δ = 0:
16n² − 20(n² − 5) = 0
−4n² + 100 = 0 → n² = 25 → n = ±5

Tangents: y = 2x + 5 and y = 2x − 5.

🎯 Quick formula:
For circle x² + y² = r² and slope m, tangents are y = mx ± r√(1 + m²).
Here r = √5, m = 2 → y = 2x ± √5·√5 = 2x ± 5 ✓

💡 Always two tangent lines for a given slope — one above and one below.
Question 6
10.00 pts

🎯 Shifted circle:

Find the horizontal tangent lines to the circle \((x-1)^2+(y-2)^2=4\).

Explanation:
💡 Detailed Explanation:

Horizontal tangents to a shifted circle! 🎯

📐 Centre and radius:
(x − 1)² + (y − 2)² = 4 → centre (1, 2), radius 2.

🎯 Horizontal tangents:
y = b ± r = 2 ± 2 → y = 4 and y = 0.

📊 Verify: distance from (1, 2) to y = 4 is 2 ✓; distance to y = 0 is 2 ✓

💡 Geometric picture: the topmost point of the circle is (1, 4) and the bottommost is (1, 0). The horizontal lines through these are exactly the horizontal tangents.
Question 7
10.00 pts

🔢 How many tangents:

How many horizontal tangents does a circle have?

Explanation:
💡 Detailed Explanation:

Always exactly two horizontal tangents! 🔢

📐 Why two:
A horizontal line has form y = c. It is tangent to the circle iff its perpendicular distance to the centre equals the radius. Exactly two horizontal lines lie at distance r from the centre — one above, one below.

🎯 Formula:
For circle (x − a)² + (y − b)² = r²: horizontal tangents are y = b + r and y = b − r.

💡 By symmetry: there are always exactly 2 vertical tangents too — at the leftmost and rightmost points. More generally, for any slope m, there are exactly 2 tangent lines.
Question 8
10.00 pts

📍 External point:

How many tangents to the circle \(x^2+y^2=4\) pass through the point \((4,0)\)?

Explanation:
💡 Detailed Explanation:

Tangents from an external point! 📍

📐 Position: circle centre (0, 0), radius 2. Distance to (4, 0) is 4 > 2 → external point.

🎯 Three cases for a point P:
• Inside the circle → 0 tangents through P
• On the circle → 1 tangent (the unique tangent at that point)
• Outside the circle → 2 tangents

📊 Geometric picture:
From an external point you can "see" the circle from two opposite sides; each visible edge gives a tangent line. The two tangent segments from an external point have equal length.

💡 Length of tangent segments from (4, 0):
Pythagoras on the (centre, P, tangent point) right triangle:
√(4² − 2²) = √12 = 2√3 ≈ 3.46.
Question 9
10.00 pts

⚠️ Mistake:

A student found the tangent \(y=3x+10\) to the circle \(x^2+y^2=1\). Is this possible?

Explanation:
💡 Detailed Explanation:

Tangent test: distance vs. radius! ⚠️

📐 Compute the distance:
Line y = 3x + 10 → 3x − y + 10 = 0 (A=3, B=−1, C=10).
Centre (0, 0):
d = |10| / √(9 + 1) = 10/√10 = √10 ≈ 3.16

Radius r = 1. Compare: d ≈ 3.16 ≫ 1 → not tangent.

🎯 Tangent formula sanity-check:
For x² + y² = r² and slope m, valid tangents are y = mx ± r√(1 + m²).
r = 1, m = 3 → y = 3x ± √10 ≈ 3x ± 3.16, not ±10.

💡 Always sanity-check: compute the centre-to-line distance and compare with r before declaring a line "tangent".
Question 10
10.00 pts

📚 Summary:

What is true about a tangent line to a circle?

Explanation:
💡 Detailed Explanation:

Tangent-line summary! 📚

🎯 Three equivalent ways to characterise a tangent:

1️⃣ Single intersection point with the circle.
2️⃣ Perpendicular to the radius at the point of contact.
3️⃣ Distance from centre = r (equivalently, Δ = 0 in the substitution-quadratic).

📐 Useful tangent formulas:
• Horizontal tangents: y = b ± r
• Vertical tangents: x = a ± r
• Tangent at point (x₀, y₀) on x² + y² = r²: x₀·x + y₀·y = r²
• Tangent with slope m to x² + y² = r²: y = mx ± r√(1 + m²)

📊 Counting tangents from an external point:
Always 2. From a point on the circle: 1. From inside: 0.

💡 Key insight: tangency is a "borderline" condition between secant (cutting through) and missing entirely. That''s exactly what Δ = 0 captures algebraically.