Analytical Geometry: Inclination, Circles & Tangents

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In Grade 10, you used distance, midpoint, and gradient to work with straight lines. In Grade 11, two major new ideas arrive: the angle of inclination (connecting gradients to trigonometry) and the equation of a circle (bringing curves into analytical geometry for the first time).

The Angle of Inclination

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The angle of inclination ($\theta$) is the angle a line makes with the positive $x$-axis, measured anti-clockwise.

⚠️ The obtuse angle trap: If $m < 0$, your calculator gives a negative angle (e.g., $-45°$). You must add $180°$ to get the correct inclination (e.g., $135°$).

Angle Between Two Lines

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If two lines have inclinations $\theta_1$ and $\theta_2$, the acute angle between them is:

The Equation of a Circle

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A circle with centre $(a;\, b)$ and radius $r$ has the equation:

Special case: Centre at the origin → $x^2 + y^2 = r^2$.

Converting General Form to Standard Form

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Sometimes the equation is given as $x^2 + y^2 + Dx + Ey + F = 0$. To find the centre and radius, complete the square:

1. Group $x$-terms and $y$-terms: $(x^2 + Dx) + (y^2 + Ey) = -F$
2. Complete the square for each group
3. Read off the centre $(a;\, b)$ and radius $r$

Example: $x^2 + y^2 - 6x + 4y - 12 = 0$

$(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$

$(x - 3)^2 + (y + 2)^2 = 25$

Centre: $(3;\, -2)$. Radius: $\sqrt{25} = 5$.

Tangent to a Circle

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A tangent touches the circle at exactly one point. The key fact:

To find the tangent equation:

1. Find the gradient of the radius from the centre to the point of tangency.
2. The tangent gradient is the negative reciprocal: $m_t = -\frac{1}{m_r}$
3. Use point-gradient form: $y - y_1 = m_t(x - x_1)$

Deep Dive

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• Inclination, Circles & Tangent Lines — full worked examples for angle of inclination, equation of a circle, completing the square, and finding tangent equations

🚨 Common Mistakes

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1. Obtuse angle of inclination: If $m < 0$, add $180°$ to the calculator answer. Don’t give a negative angle.
2. Signs in the circle equation: $(x - 3)^2 + (y + 2)^2 = 25$ has centre $(3;\, -2)$, NOT $(3;\, 2)$. The signs flip!
3. Completing the square: Add the “completing” values to BOTH sides of the equation.
4. Tangent gradient: It’s the negative reciprocal of the radius gradient, not just the reciprocal.
5. Not verifying the point is on the circle: Before finding a tangent, substitute the point into the circle equation to confirm.

🔗 Related Grade 11 topics:

• Trigonometry: Reduction — $\tan\theta = m$ connects gradients to angles
• Circle Geometry — the geometric approach to circles complements the algebraic approach here
• Quadratic Equations — completing the square is used for circle equations

📌 Grade 10 foundation: Analytical Geometry: Core Formulas

📌 Grade 12 extension: Analytical Geometry — more complex tangent problems and circle intersections

⏮️ Circle Geometry | 🏠 Back to Grade 11 | ⏭️ Statistics