Problem 1: what are conic sections or conics?
Ans: consider the double cone of figure 1. Some of the cross sections (plane slices) that can be obtained from the double cone are:
(a) Single point
(b) Single line
(c) Double lines
These shapes are called conic sections or conics. It will be shown later in the ellipse problems that the circle is a special case of an ellipse. The parabola, ellipse and hyperbola are called non-degenerate conics while the single point, single line and double lines are called degenerate conics
Problem 2: the equation of a circle in R2 is given as:
(x - a)2 + (y - b)2 = r2
Determine the coordinates of the center of the circle and the length of its radius.
Ans: center of circle is at (a,b); radius is r units.
Problem 3: consider the following curve in R2:
x2 + y2 - 8x -10y + 5 = 0.
Is this curve a circle? if yes, determine the coordinates of its center and the length of its radius.
Ans: reduce the equation of the curve to the equation of a circle as represented in problem 2 by completing the squares:
(x - 4)2 + (y - 5)2 -16 -25 + 5 = 0
therefore, (x - 4)2 + (y - 5)2 = 62
therefore, curve is a circle with center at (4,5) and radius of 6 units
In general, an equation of the form:
x2 + y2 + fx + gy + h = 0
represents a circle with
center (-f/2, -g/2) and radius √(f2/4 + g2/4 - h).
so long as f2/4 + g2/4 - h > 0.
Problem 4 : Determine the set of points (x,y) in R2 that satisfies the following equations:
(a) x2 + y2 + x/2 + y/2 + 1 = 0.
(b) x2 + y2 - 4x + 8y + 20 = 0.
(c) 2x2 + 2y2 - 8x -10y + 5 = 0.
Ans: (a) f2/4 + g2/4 - h < 0, therefore no points (x,y) satisfy the equation.
(b) f2/4 + g2/4 - h = 0, therefore the equation represents a single point.
(c) Divide equation by 2. Then f = -4, g = -5 and h = 5/2.
f2/4 + g2/4 - h > 0, therefore,
equation represents a circle with center at (-f/2, -g/2) = (2,5/2) and radius is √ (f2/4 + g2/4 - h) = √(41)/2.
Problem 5 : C1 and C2 are mutually orthogonal intersecting circles with the following equations:
C1: x2 + y2 - 4x - g1y + 7 = 0
C2: x2 + y2 + 2x -8y + 5 = 0
Determine the value of g1.
Ans: Two intersecting circles are mutually orthogonal if and only if:
f1f2 + g1g2 = 2(h1 + h2)
Therefore, -8 + 8g1 = 24
Therefore, g1 = 4 and -g1 = -4
Problem 6: C1 and C2 are circles that intersect at distinct points P and Q.
C1: x2 + y2 + f1x + g1y + h1 = 0
C2: x2 + y2 + f2x + g2y + h2 = 0
What is the general form of the equation of the line and any other circle through P and Q?
Ans: the general form of the equation is as follows:
x2 + y2 + f1x + g1y + h1 + k(x2 + y2 + f2x + g2y + h2) = 0 --- (1); for some number k.
If k ≠ -1, this equation is one of the circles. If k = -1, then it is the equation of the line.
Problem 7: consider circles C1, C2 and equation (1) of problem 6.
what value of k will reduce equation (1) to the equation of circle C2?
What does this value imply with respect to equation (1) of problem 6?
Ans: k = ∞. Divide equation (1) of problem 6 by k and let k ----> ∞. The implication is that C2 is excluded from the group of circles represented by equation (1) of problem 6.
Problem 8:These two circles intersect at distinct points P and Q:
C1: 2x2 + 2y2 - 6x + 8y - 2 = 0
C2: 2x2 + 2y2 + 5x - 6y + 3 = 0
Determine the equation of the circle that passes through (1,0) and the points P and Q.
Ans: this equation is of the form:
x2 + y2 - 3x + 4y - 1 + k(x2 + y2 + (5/2)x - 3y + 3/2) = 0
If it passes through (1,0), then 1 - 3 -1 + k(1 + 5/2 +3/2) = 0. Therefore, k = 3/5.
Therefore, the equation of the circle is:
x2 + y2 - 3x + 4y - 1 + 3/5(x2 + y2 + (5/2)x - 3y + 3/2) = 0
which simplifies to:
16x2 + 16y2 - 15x + 22y - 1 = 0.
Problem 9: Determine the equation of the line that passes through the points of intersection of circles C1 and C2 in problem 8.
Ans: k = -1 in this case. Therefore the equation is:
x2 + y2 - 3x + 4y - 1 -1(x2 + y2 + (5/2)x - 3y + 3/2) = 0
11x - 14y + 5 = 0.
Problem 10: consider circles C1 and C2:
C1: x2 + y2 + 4x + 4y + 8 = 0
C2: x2 + y2 + 6x + 2y + 4 = 0
(a) What points in R2 satisfy the result obtained when C2 is subtracted from C1?
(b) Now consider the following two circles:
C3: x2 + y2 = 1
C4: x2 + y2 = 4
What points in R2 satisfy the result obtained when C3 is subtracted from C4?
Ans: No help here :).
Peter Oye Sagay