L27-L28: Geometry I — Circle & Parabola

Lecture notes covering the geometry of circles and parabolas, including equations, intersections, tangents, normals, and key properties.

Learning Outcomes

  1. Determine the equation of a circle.
  2. Determine the centre and radius of a circle.
  3. Find points of intersection of two circles, and of a circle and a line.
  4. Find the equations of tangent and normal lines to a circle.
  5. Find the length of a tangent from a point to a circle.
  6. Determine the equation of a parabola given its vertex and focus.
  7. Determine the vertex, focus, and equation of a parabola by completing the square.
flowchart TD
    A([Conic Sections]) --> B[Circle]
    A --> C[Parabola]
    B --> B1["Standard Eq:<br/>(x-h)² + (y-k)² = r²"]
    B --> B2["General Eq:<br/>x² + y² + Dx + Ey + F = 0"]
    B --> B3["Intersection with Line<br/>Δ > 0: two points<br/>Δ = 0: tangent<br/>Δ < 0: none"]
    B --> B4["Tangent & Normal<br/>⊥ to radius at point"]
    B --> B5["Length of Tangent<br/>from (m,n):<br/>√[(m-h)²+(n-k)²-r²]"]
    C --> C1["Definition:<br/>equidistant from focus & directrix"]
    C --> C2["Standard Forms:<br/>(x-h)² = 4a(y-k) vertical<br/>(y-k)² = 4a(x-h) horizontal"]
    C --> C3["Key Elements:<br/>Vertex, Focus, Directrix<br/>Axis, Latus Rectum = 4a"]

Circle

Definition

A circle is the curve consisting of all points $P$ in a plane that are equidistant from a fixed point (the centre). The radius is the fixed distance from the centre to the curve.

If the distance between $P(x,y)$ and centre $C(h,k)$ is constant and equal to $r$: $$CP = r \implies \sqrt{(x-h)^2 + (y-k)^2} = r$$ $$(CP)^2 = (x-h)^2 + (y-k)^2 = r^2$$

Standard Equation

For a circle with centre $(h,k)$ and radius $r$: $$(x-h)^2 + (y-k)^2 = r^2$$

Special case (centre at origin): $$x^2 + y^2 = r^2$$

General Equation

Expanding the standard equation: $$(x-h)^2 + (y-k)^2 = r^2$$ $$\implies x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2$$ $$\implies x^2 - 2hx + y^2 - 2ky + (h^2 + k^2 - r^2) = 0$$

Let $C = h^2 + k^2 - r^2$. Then: $$x^2 - 2hx + y^2 - 2ky + C = 0 \quad ; \quad C = h^2 + k^2 - r^2$$

Radius formula: $$r = \sqrt{h^2 + k^2 - C}$$

Intersection of Circle and Straight Line

The intersection points are found by solving simultaneous equations.

Condition Roots Geometric Meaning
Two distinct real roots $\Delta > 0$ Line intersects circle at two points $P$ and $Q$
One repeated real root $\Delta = 0$ Line is tangent to the circle at $P$
No real roots $\Delta < 0$ Line does not intersect the circle

Tangent and Normal to a Circle

  • The tangent at a point on a circle is perpendicular to the radius at that point.
  • The normal at a point is perpendicular to the tangent (and thus passes through the centre).

Length of Tangent from an External Point

For external point $T(m,n)$, tangent point $S$, and centre $C(h,k)$: $$ST = \sqrt{(CT)^2 - (CS)^2}$$ $$ST = \sqrt{(m-h)^2 + (n-k)^2 - r^2}$$

Examples from Lecture

Example 1: Find the equation of circle with a) centre $(2,-3)$ and radius $5$ b) centre origin and radius $2$

Example 2: Find the centre and radius of the circle a) $x^2 + y^2 + 4x - 6y - 23 = 0$ b) $x^2 + y^2 + 5x - 6y - 5 = 0$ c) $2x^2 + 2y^2 - 8x + 6y + 5 = 0$

Example 3: Find the equation of circle with centre $(-1,2)$ which touches the line $4x - 3y = 10$.

Example 4:

  1. Find the equation of circle passing through $A(1,8)$, $B(-6,1)$, $C(-2,-1)$.
  2. Find the equation of circle passing through $A(1,3)$ and $B(-1,-1)$ with diameter on $x + 2y = 1$.

Example 5: Determine the point of intersections of $x^2 + y^2 = 4$ and $x^2 + y^2 - 2x + 4y + 4 = 0$.

Example 6:

  1. Find intersection of line $4x - 3y + 1 = 0$ with circle $x^2 + y^2 - 4x - 6y - 12 = 0$.
  2. Determine $k$ if $x^2 + y^2 - 2x + 6y + k = 0$ touches $3x + 4y = 16$. Find point of contact.
  3. Show $3y - 4x - 42 = 0$ does not intersect $x^2 + y^2 + 4x - 6y - 9 = 0$. Find shortest distance.

Example 7:

  1. Find equation of tangent to $x^2 + y^2 + 6x + y - 7 = 0$ at $P(-1,3)$.
  2. Find equation of normal to $x^2 + y^2 - 4x + 4y - 2 = 0$ at $P(5,-3)$.

Example 8: Find lengths of tangents from: a) $(3,5)$ to $x^2 + y^2 + 2x - 4y - 4 = 0$ b) $(5,-1)$ to $x^2 + y^2 - x - 6 = 0$ c) $(2,-3)$ to $2x^2 + 2y^2 - 3x + y + 1 = 0$

Parabola

Definition

A parabola is the curve consisting of all points $P$ in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

Derivation

When vertex is at origin, focus $F(0,a)$, directrix $y = -a$: $$\sqrt{x^2 + (y-a)^2} = y + a$$ $$x^2 + (y-a)^2 = (y+a)^2$$ $$x^2 = 4ay$$

When vertex is at $(h,k)$, focus $F(h, k+a)$, directrix $y = k-a$: $$\sqrt{(x-h)^2 + (y-(k+a))^2} = |y-(k-a)|$$ $$(x-h)^2 = 4a(y-k)$$

Important Terms

  • Focus: fixed point, $a$ units from vertex on the axis
  • Directrix: fixed line perpendicular to axis, $a$ units from vertex
  • Axis: line through focus and vertex, perpendicular to directrix
  • Vertex: point of intersection between parabola and axis; midpoint of focus and directrix
  • Latus rectum: chord through the focus parallel to the directrix

For $x^2 = 4ay$, when $y = a$, points are $(\pm 2a, a)$. Length of latus rectum: $$\sqrt{(2a - (-2a))^2 + (a-a)^2} = 4a$$

Standard Equations

Orientation Equation Vertex Focus Directrix Shape
Vertical $(x-h)^2 = 4a(y-k); a>0$ $(h,k)$ $(h, k+a)$ $y = k-a$ Opens upward
Vertical $(x-h)^2 = 4a(y-k); a<0$ $(h,k)$ $(h, k-a)$ $y = k+a$ Opens downward
Horizontal $(y-k)^2 = 4a(x-h); a>0$ $(h,k)$ $(h+a, k)$ $x = h-a$ Opens to the right
Horizontal $(y-k)^2 = 4a(x-h); a<0$ $(h,k)$ $(h-a, k)$ $x = h+a$ Opens to the left

Examples from Lecture

Example 9: Find equations of parabola with: a) Vertex $(0,0)$, Focus $(2,0)$ b) Vertex $(0,0)$, Focus $(0,-2)$ c) Vertex $(3,2)$, Focus $(4,2)$ d) Vertex $(-4,3)$, Focus $(-4,1)$ Then sketch.

Example 10: Find focus, vertex, directrix: a) $x^2 = 16y$ b) $y^2 = -2x$ c) $(x+4)^2 = -20y - 20$ d) $y^2 + 6y + 1 + 4x = 0$ Then sketch.

Example 11: Determine equation of parabola with axis parallel to $y$-axis, vertex at $(2,-1)$, passing through $(3,1)$.

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