Specific Expectations

MCR3U.4.1 Representing Loci

By the end of this course, students will:

• MCR3U.4.1.a construct a geometric model (e.g., a diagram created by hand, a diagram created by using dynamic geometry software) to represent a described locus of points;
• MCR3U.4.1.b determine the properties of the geometric model; and use the properties to interpret the locus (e.g., the locus of points  equidistant from two fixed points is the right bisector of the line segment joining the two fixed points);
• MCR3U.4.1.c explain the process used in constructing a geometric model of a described locus;
• MCR3U.4.1.d determine an equation to represent a described locus [e.g., determine the equation of the locus of points equidistant from (–2, 7) and (5, 4)];
• MCR3U.4.1.e construct geometric models to represent the locus definitions of the conics;
• MCR3U.4.1.f determine equations for conics from their locus definitions, by hand for simple particular cases [e.g., determine the equation of the locus of points the sum of whose distances from (–3, 0) and (3, 0) is 10].

MCR3U.4.2 Determining the Equation and the Key Features of a Conic

By the end of this course, students will:

• MCR3U.4.2.a identify the standard forms for the equations of parabolas, circles, ellipses, and hyperbolas having centres at (0, 0) and at (h, k);
• MCR3U.4.2.b identify the type of conic, given its equation in the form ax² + by² + 2gx + 2fy + c = 0;
• MCR3U.4.2.c determine the key features (e.g., the centre or the vertex, the focus or foci, the asymptotes, the lengths of the axes) of a conic whose equation is given in the form ax² + by² + 2gx + 2fy + c = 0, by hand in simple cases (e.g., x² + 9y² – 6x + 36y – 36 = 0);
• MCR3U.4.2.d sketch the graph of a conic whose equation is given in the form ax2 + by 2 + 2gx + 2fy + c = 0;
• MCR3U.4.2.e illustrate the conics as intersections of planes with cones, using concrete materials or technology.

MCR3U.4.3 Solving Problems Involving Applications of the Conics

By the end of this course, students will:

• MCR3U.4.3.a describe the importance, within applications, of the focus of a parabola, an ellipse, or a hyperbola (e.g., all incoming rays parallel to the axis of a parabolic antenna are eflected through the focus; the planets move in elliptical orbits with the sun at one of the foci);
• MCR3U.4.3.b pose and solve problems drawn from a variety of applications involving conics, and communicate the solutions with clarity and justification (Sample problem: A parabolic antenna is 320 m wide at a distance of 50 m above its vertex. Determine the distance above the vertex of the focus of the antenna);
• MCR3U.4.3.c solve problems involving the intersections f lines and conics.