SECTION 2.2. Distance and Midpoint Formulas; Circles


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1 SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation is in standard form.. Convert the general form of a circle s equation to standard form. Distance and Midpoint Formulas; Circles Section. Distance and Midpoint Formulas; Circles 9 It s a good idea to know our wa around a circle. Clocks, angles, maps, and compasses are based on circles. Circles occur everwhere in nature: in ripples on water, patterns on a butterfl s wings, and cross sections of trees. Some consider the circle to be the most pleasing of all shapes. The rectangular coordinate sstem gives us a unique wa of knowing a circle. It enables us to translate a circle s geometric definition into an algebraic equation. To do this, we must first develop a formula for the distance between an two points in rectangular coordinates. Find the distance between two points. The Distance Formula Using the Pthagorean Theorem, we can find the distance between the two points P (, ) and P (, ) in the rectangular coordinate sstem. The two points are illustrated in Figure.. P (, ) P (, ) d (, ) Figure. The distance that we need to find is represented b d and shown in blue. Notice that the distance between two points on the dashed horizontal line is the absolute value of the difference between the coordinates of the two points. This distance, , is shown in pink. Similarl, the distance between two points on the dashed vertical line is the absolute value of the difference between the coordinates of the two points. This distance, , is also shown in pink.
2 9 Chapter Functions and Graphs P (, ) Because the dashed lines are horizontal and vertical, a right triangle is formed. Thus, we can use the Pthagorean Theorem to find distance d. B the Pthagorean Theorem, P (, ) d (, ) d = d = d = (  ) + (  ). Figure., repeated This result is called the distance formula. The Distance Formula The distance, d, between the points (, ) and (, ) in the rectangular coordinate sstem is d = (  ) + (  ). When using the distance formula, it does not matter which point ou call (, ) and which ou call (, ). EXAMPLE Using the Distance Formula Find the distance between (, ) and (, ). Solution Letting (, ) = (, ) and (, ) = (, ), we obtain d = (  ) + (  ) Use the distance formula. (, ) = [  ()] + [  ()] = ( + ) + ( + ) = + = 9 + Substitute the given values. Appl the definition of subtraction within the grouping smbols. Perform the resulting additions. Square and. (, ) Distance is units. = = L.7. Add. = 9 = 9 = Figure. Finding the distance between two points The distance between the given points is units, or approimatel.7 units. The situation is illustrated in Figure.. Find the distance between (, ) and (, ). Find the midpoint of a line segment. The Midpoint Formula The distance formula can be used to derive a formula for finding the midpoint of a line segment between two given points. The formula is given as follows:
3 Section. Distance and Midpoint Formulas; Circles 9 The Midpoint Formula Consider a line segment whose endpoints are (, ) and (, ). The coordinates of the segment s midpoint are +, +. To find the midpoint, take the average of the two coordinates and the average of the two coordinates ( 8, ) 7 (, ) Midpoint (, ) EXAMPLE Using the Midpoint Formula Find the midpoint of the line segment with endpoints (, ) and (8, ). Solution To find the coordinates of the midpoint, we average the coordinates of the endpoints. Midpoint = + (8) Average the coordinates.  + (), Average the coordinates. = 7, 0 =  7,  Figure. Finding a line segment s midpoint Figure. illustrates that the point (, ) and (8, ). ( 7, ) is midwa between the points Find the midpoint of the line segment with endpoints (, ) and (7, ). Circles Our goal is to translate a circle s geometric definition into an equation. We begin with this geometric definition. Center (h, k) Radius: r (, ) An point on the circle Figure.7 A circle centered at (h, k) with radius r Definition of a Circle A circle is the set of all points in a plane that are equidistant from a fied point, called the center. The fied distance from the circle s center to an point on the circle is called the radius. Figure.7 is our starting point for obtaining a circle s equation. We ve placed the circle into a rectangular coordinate sstem. The circle s center is (h, k) and its radius is r. We let (, ) represent the coordinates of an point on the circle. What does the geometric definition of a circle tell us about point (, ) in Figure.7? The point is on the circle if and onl if its distance from the center is r. We can use the distance formula to epress this idea algebraicall: The distance between (, ) and (h, k) (  h) + (  k) is alwas r. = r
4 9 Chapter Functions and Graphs Squaring both sides of (  h) + (  k) = r ields the standard form of the equation of a circle. The Standard Form of the Equation of a Circle The standard form of the equation of a circle with center (h, k) and radius r is (  h) + (  k) = r. Write the standard form of a circle s equation. (0, 0) Figure.8 The graph of + = (, ) + = EXAMPLE Finding the Standard Form of a Circle s Equation Write the standard form of the equation of the circle with center (0, 0) and radius. Graph the circle. Solution The center is (0, 0). Because the center is represented as (h, k) in the standard form of the equation, h = 0 and k = 0. The radius is, so we will let r = in the equation. (  h) + (  k) = r (  0) + (  0) = + = This is the standard form of a circle s equation. Substitute 0 for h, 0 for k, and for r. Simplif. The standard form of the equation of the circle is + =. Figure.8 shows the graph. Technolog = Write the standard form of the equation of the circle with center (0, 0) and radius. To graph a circle with a graphing utilit, first solve the equation for. + = =  Graph the two equations =;  =  and =  = in the same viewing rectangle. The graph of =  is the top semicircle because is alwas positive.the graph of =  is the bottom semicircle because is alwas negative. Use a ZOOM SQUARE setting so that the circle looks like a circle. (Man graphing utilities have problems connecting the two semicircles because the segments directl across horizontall from the center become nearl vertical.) Eample and involved circles centered at the origin. The standard form of the equation of all such circles is + = r, where r is the circle s radius. Now, let s consider a circle whose center is not at the origin.
5 EXAMPLE Section. Distance and Midpoint Formulas; Circles 97 Finding the Standard Form of a Circle s Equation Write the standard form of the equation of the circle with center (, ) and radius. Solution The center is (, ). Because the center is represented as (h, k) in the standard form of the equation, h = and k =. The radius is, so we will let r = in the equation. (  h) + (  k) = r This is the standard form of a circle s equation. [  ()] + (  ) = Substitute for h, for k, and for r. ( + ) + (  ) = Simplif. The standard form of the equation of the circle is ( + ) + (  ) =. Write the standard form of the equation of the circle with center (, ) and radius 0. Give the center and radius of a circle whose equation is in standard form. EXAMPLE Using the Standard Form of a Circle s Equation to Graph the Circle Find the center and radius of the circle whose equation is and graph the equation. (  ) + ( + ) = 9 Solution In order to graph the circle, we need to know its center, (h, k), and its radius, r. We can find the values for h, k, and r b comparing the given equation to the standard form of the equation of a circle. (, ) 7 (, ) 7 (, ) (, 7) Figure.9 The graph of (  ) + ( + ) = 9 (, ) (  ) + [  ()] = This is ( h), with h =. (  ) + ( + ) = 9 This is ( k), with k =. This is r, with r =. We see that h =, k =, and r =. Thus, the circle has center (h, k) = (, ) and a radius of units. To graph this circle, first plot the center (, ). Because the radius is, ou can locate at least four points on the circle b going out three units to the right, to the left, up, and down from the center. The points three units to the right and to the left of (, ) are (, ) and (, ), respectivel. The points three units up and down from (, ) are (, ) and (, 7), respectivel. Using these points, we obtain the graph in Figure.9. Find the center and radius of the circle whose equation is and graph the equation. ( + ) + (  ) =
6 98 Chapter Functions and Graphs If we square  and + in the standard form of the equation from Eample, we obtain another form for the circle s equation. (  ) + ( + ) = 9 This is the standard form of the equation from Eample = 9 Square  and = 9 Combine numerical terms and rearrange terms = 0 Subtract 9 from both sides. This result suggests that an equation in the form + + D + E + F = 0 can represent a circle. This is called the general form of the equation of a circle. The General Form of the Equation of a Circle The general form of the equation of a circle is + + D + E + F = 0. Convert the general form of a circle s equation to standard form. Stud Tip To review completing the square, see Section., pages 9 0. (, ) Figure.0 The graph of ( + ) + (  ) = We can convert the general form of the equation of a circle to the standard form (  h) + (  k) = r. We do so b completing the square on and. Let s see how this is done. EXAMPLE Write in standard form and graph: Converting the General Form of a Circle s Equation to Standard Form and Graphing the Circle Solution Because we plan to complete the square on both and, let s rearrange terms so that terms are arranged in descending order, terms are arranged in descending order, and the constant term appears on the right = 0 ( + ) + (  ) = ( + + ) + (  + 9) = Remember that numbers added on the left side must also be added on the right side. ( + ) + (  ) = This is the given equation. Rewrite in anticipation of completing the square. Complete the square on : = and =, so add to both sides. Complete the square on : and () () = = 9, so add 9 to both sides. Factor on the left and add on the right. This last equation is in standard form. We can identif the circle s center and radius b comparing this equation to the standard form of the equation of a circle, (  h) + (  k) = r. [  ()] + (  ) = This is ( h), with h = = 0. ( + ) + (  ) = This is ( k), with k =. This is r, with r =. We use the center, (h, k) = (, ), and the radius, r =, to graph the circle. The graph is shown in Figure.0.
7 Technolog Eercise Set. 99 To graph = 0, rewrite the equation as a quadratic equation in. Now solve for using the quadratic formula, with a =, b =, and c = + . = b ; b  ac a Because we will enter these equations, there is no need to simplif. Enter = +  A +  B  + ( +  ) = 0 = () ; ()  A +  B = ;  A +  B and Use a ZOOM SQUARE =   A +  B. setting. The graph is shown on the right. Write in standard form and graph: = 0.
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