Straight Lines and Circles

$(x-a)^2 + (y-b)^2 = r^2$. Centre $(a,b)$, radius $r$.

General form: $x^2 + y^2 + 2gx + 2fy + c = 0$. Centre $(-g, -f)$, radius $\sqrt{g^2 + f^2 - c}$.

• Tangent at point $P$: perpendicular to radius $OP$.
• Normal at point $P$: passes through centre $O$ and $P$.

Substitute the line into the circle equation. Discriminant determines:

• $\Delta > 0$: two intersection points (secant)
• $\Delta = 0$: tangent
• $\Delta < 0$: no intersection

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

Centre: $(3, -2)$. Radius: $\sqrt{9 + 4 + 12} = 5$.

Tangent at $(6, 2)$: gradient of radius = $\frac{2-(-2)}{6-3} = \frac{4}{3}$. Tangent gradient = $-\frac{3}{4}$.

$y - 2 = -\frac{3}{4}(x - 6)$ → $y = -\frac{3}{4}x + \frac{13}{2}$.