Calculus: Analytic Geometry and Calculus, with Vectors

30 Analytic geometry in two dimensions

and that the equation of the tangent to C at a pointPi(xi,yi) onC can be put in

the form

(xi - xo)(x - xo) + (Yi - yo)(y - Yo) = a2.

Hint: Draw a figure and notice that when x, 0 xo, we cancalculate the slope of

the line PoP1 and use the fact (from plane geometry)that the tangent to C at

Pi is perpendicular to PoPI.

9 Show that the circle passing through the three pointsA(0,2), B(2,0), and

C(4,0) has its center at the point (3,3) and has radius. Hint: The per-

pendicular bisectors of the segments AB and BC are easily found, and their

intersection is the required center.

10 A circle passes through the points (0,7) and (0,9) and is tangent to the

x axis at a point on the negative x axis. Find the radius, center, and equation

of the circle.

11 Let 0 < a < b and find the radius r and center (h,k) of the circle which

passes through the points (0,a) and (0,b) andwhich is tangent to the x axis at a

point to the left of the origin. Ins.:

a + b a + b

r = 2 , h=-, k=^2

12 A circle has a diameter (line segment, not number) on the x axis. The

circle contains the two points (a,0) and (b,c) for which c 0 0. Show that b 0 a

and find the center of the circle. Ins.:

/b2 + c2- a2

2(b- a) , 0

13 The points 11(- a,0) and B(a,0) are the ends of a diameter (line segment,

not number) of a circle of radius a having its center at the origin. Write and

simplify the equation which x and y must satisfy if A, B, and P(x,y) are vertices

of a right triangle the side AB of which is the hypotenuse.

14 An equilateral triangle has its center at the origin and has one vertex

at the point (a,0). Find the coordinates of the other vertices and check the

results by use of the distance formula.

15 Sketch a figure which shows whether there are values of y for which the

point (O,y) is equidistant from the points (-4,1) and (7,-2). Then attack the

problem analytically. Make everything check.

16 An equilateral triangle in the closed first quadrant has vertices at the

origin and at (a,0). Find the coordinates of the third vertex and the slopes of

the sides.

17 An isosceles triangle is placed upon a coordinate system in such a way

that its vertices are (-a,0) ,(a,0), and (0,b). Prove analytically that two of the

medians have equal lengths.

18 A triangle has vertices at A(-a,0), B(b,0), C(0,c). Prove that if the

medians drawn from A and B have equal lengths, then the triangle is isosceles.

19 Find the values of the constant b for which the line having the equation

y = 2x + b intersects the circle having the equation x2 + y2 = 25. Ins.:

IbI 5 -vrl-25.