#$&* course mth 164 3:30pm 11/21 012. Equations and Properties of Parabolas, hyperbolas and ellipses*********************************************
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Given Solution: `aFigure 74 shows the point (7,4), now identified as the focus, and the line y = 2, now identified as the directrix. Recall that in the previous exercise set we located points which were equidistant from this line and point. These points are again shown, along with the curve (a parabola) consisting of all such points. The vertex is the point halfway between the focus and the directrix; this point clearly has coordinate form (7, 3), lying 1 unit lower than the focus and one unit higher than the directrix. We often use a to represent the displacement from vertex to focus, or directrix to vertex. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q002. The equation (y - k) = 1 / (4 a) * ( x - h )^2 describes the parabola with vertex (h, k) and displacement a from vertex to focus. Give the equation of the parabola from the preceding problem whose focus is at (7, 4) and whose directrix is at y = -2. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I am confused because the book teaches a different equation for this, they teach that if it opens up the equation would be (x - h)^2 = 4a(y - k) which in this case would be (x - 7)^2 = 4(1)(y - 3) is this just a different way to get the same answer? confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe vertex is at (h, k) = (7, 3) and as seen in the preceding solution we have a = 1. Thus the equation is (y - 3) = 1 / (4 * 1) * (x - 7)^2, which simplifies to y - 3 = 1/4 * (x - 7)^2. Note that this is identical with the solution obtained in the preceding exercise, where we simplified the equation sqrt( (x-7)^2 + (y-4)^2 ) = y - 2, representing the statement that the distance from (7,4) to (x,y) is equal to the distance from (x,y) to the line y = 2. After several steps of simplification we obtained the form (y - 3) = 1/4 ( x - 7)^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating:
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Given Solution: `aFigure 35 illustrates the parabola, whose vertex lies 1 unit lower than the focus and whose directrix must therefore lie 1 unit lower than the vertex (recall that the displacement between vertex and focus is the same as the displacement from the directrix to the vertex). The directrix is therefore the line y = 2. The equation of the parabola is (y - k) = 1 / (4 a) * (x - h)^2, where (h,k) is the coordinate form of the vertex and a is the displacement from vertex to focus. Thus (h, k) = (7, 3) so h = 7 and k = 3, while a is the -1 unit displacement from vertex to focus. The equation of the parabola is therefore y - 3 = 1 / (4 * -1) * (x - 7)^2, or y - 3 = -1/4 * (x-7)^2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q004. The set of points the sum of whose distances from the two given points (0, c) and (0, -c) is equal to 2b is called an ellipse. Its equation is x^2 / a^2 + y^2 / b^2 = 1, where a^2 = b^2 - c^2. The points (0, -c) and (0, c) are called the foci of the ellipse; each individual point is called a focus. What is the equation of the ellipse with foci at (0, -3) and (0, 3) for which a = 4? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: x^2/16 + y^2/7 = 1 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe coordinates of the foci are (-c, 0) and (c, 0), identified here as (-3,0) and (3,0). So we know that c = 3. Since a = 4 and a^2 = b^2 - c^2 we have b = sqrt(a^2 + c^2) = sqrt(4^2 + 3^2) = sqrt(25) = 5. So the equation of the given ellipse is x^2 / a^2 + y^2 / b^2 = 1, with a = 4 and b = 5, or x^2 / 4^2 + y^2 / 5^2 = 1, also written as x^2/16 + y^2/25 = 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I thought that a^2 - c^2 = b^2 at least that is what is in the textbook. Please clarify. ------------------------------------------------ Self-critique Rating: 3
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Given Solution: `aThe x intercepts of an equation lie at the points where y = 0. So the x intercepts of the equation x^2/16 + y^2/25 = 1 occur when x^2/16 + 0^2/25 = 1. We easily solve the resulting equation: x^2/16 = 1 so that x^2 = 16 and x = +- sqrt(16) = +-4. In general the equation x^2 / a^2 + y^2 / b^2 = 1 has x intercepts at x = +-a and y intercepts at y = +-b. The y intercepts of an equation lie at the points where x = 0. So the y intercepts of the equation x^2/16 + y^2/25 = 1 occur when 0^2/16 + y^2/25 = 1. We easily solve the resulting equation: y^2/25 = 1 so that y^2 = 25 and y = +- sqrt(25) = +-5. In general the equation x^2 / a^2 + y^2 / b^2 = 1 has x intercepts at x = +-a and y intercepts at y = +-b. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q006. Sketch the rectangle formed by the lines x = -4, x = 4, y = -5, and y = 5. Sketch the ellipse circumscribed by this rectangle (this will be the largest ellipse you can fit into the rectangle). What is the equation of this ellipse? What are the coordinates of its foci? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The equation for this would be x^2 / 16 + y^2 / 25 = 1 How do you find the foci ? confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe rectangle is shown in Figure 99, bounded by the vertical lines x = 4 and x = -4 and by the horizontal lines y = 5 and y = -5. The largest possible ellipse inside the rectangle is shown in Figure 33, intersecting the rectangle at the points (-4,0), (4,0), (0, -5) and (0, 5). Note that the x intercepts of this ellipse are the same as for the equation x^2 / 16 + y^2 / 25 = 1. The general method for sketching an ellipse of this form is to draw vertical lines through the x intercepts and horizontal lines through the y intercepts to form a rectangle. The ellipse can then be inscribed in the rectangle as was done here. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating:
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Given Solution: `aThe given rectangle is shown in Figure 13. The center of the rectangle will lie halfway between x = 2 and x = 10, at x = (-2 + 6) / 2 = 2, and halfway between y = -4 and y = 6, at y = (-4 + 6) / 2 = 1. The rectangle is therefore centered at (2, 1). If we shifted the rectangle so that the center was that (0, 0), we would have to shift -2 units in the x direction and -1 unit in the y direction. This would shift the lines x = -2 and x = 6 to x = -4 and x = 4, and the lines y = -4 and y = 6 to y = -5 and y = 5. This rectangle would give us the ellipse centered at the origin having intercepts at (-a, 0) = (-4, 0) and (a, 0) = (4, 0), as well as at (-b, 0) = (-5, 0) and (b, 0) = (5, 0). Thus we have a = 0 and b = 0 and the ellipse would have equation x^2 / a^2 + y^2 / b^2 = x^2 / 16 + y^2 / 25 = 0. Shifting this ellipse back to its original position with center at (h, k) = (2, 1), we replace x by x - h = x - 2 and y by y - h = y - 1 to obtain the equation (x-2)^2 / 16 + (y - 1)^2 / 25 = 1. This ellipse is shown in Figure 38. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q008. The set of points whose distances from the two given points (c, 0) and (-c, 0) differ by 2a is called a hyperbola. Its equation is x^2 / a^2 - y^2 / b^2 = 1, where c^2 = a^2 + b^2. The points (-c, 0) and (c, 0) are called the foci of the hyperbola; each individual point is called a focus. What is the equation of the hyperbola with foci at (0, -sqrt(41)) and (-, sqrt(41)) (these points are at approximately (0, +-6.4) for which a = 4? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: X^2 / 16 - y^2 / 25 = 1 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe hyperbola with foci at ( +-c, 0) = (+-sqrt(41), 0), for which a = 4, has b^2 = c^2 - a^3 = sqrt( (41)^2 - 4^2 ) = sqrt( 41 - 25) = sqrt(16) = 4, so that b = sqrt(16) = 4. So the equation of this ellipse is x^2 / a^2 - y^2 / b^2 = 1, or x^2 / 5^2 - y^2 / 4^2 = 1, which gives us x^2 / 25 - y^2 / 16 = 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I think that there is some problems with the given answer unless it is meant to be wrong. Because 4^2 does not = 25. ------------------------------------------------ Self-critique Rating:
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Given Solution: `aThe rectangle is shown in Figure 82. The diagonals are included in this figure. The hyperbola will have two branches, one starting at (4, 0) and opening to the right. This hyperbola will approach the diagonals as its asymptotes. The other branch starts at (-4, 0) and opens to the left, again approaching the diagonals as its asymptotes. The hyperbola is shown in figure 85. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q010. How does the shape of the hyperbola you just sketched differ from that of a parabola? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The shape is about the same the only difference I notice is that there are two of them instead of one. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe parabolas we have encountered so far in the course open upward or downward, while the hyperbola we see here opens to the right and to the left. This however is not really a difference between parabolas and hyperbolas, since parabolas can also open to the right or to the left (but not both on the same parabola). One difference is that a hyperbola has two branches and a parabola does not. Also the hyperbola to approaches asymptotes whereas a parabola never approaches any fixed line. A parabola which opens upward or downward keeps getting steeper and steeper, without limit. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q011. What are the equations of the diagonals of the hyperbola you sketched? How are the equations of the diagonals related to the equation of the hyperbola? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I don’t know how to answer this question because I don’t know what the equation of the diagonals are, or how they are related to the equation of the hyperbola. confidence rating #$&*: 0 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe diagonals both pass through the origin. One has a slope of 5/4 and the other a slope of -5/4, so the equations of the diagonals are y = 5/4 x and y = -5/4 x. The equation of the hyperbola is x^2 / 16 - y^2 / 25 = 1. Solving for y we have y = +-sqrt [ 25 ( x^2 / 16 + 1) ]. For very large x and y, the 1 on the right-hand side is not very significant and we therefore effectively have y = +-sqrt[ 25 x^2 / 16 ](good approximation for large x). sqrt(25 x^2 / 16) = 5 x / 4 so for large x our approximation becomes y = +- 5 x / 4, which is identical to the equations of the asymptotes. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Why are they 5/4 is it because the lines go through the corners of the rectangle that we had drawn? ------------------------------------------------ Self-critique Rating:
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Given Solution: `aFigure 51 shows the hyperbola as described. It is not completely clear from the sketch how the hyperbola approaches its asymptotes. The approach to the asymptote is illustrated for the lower branch in the subsequent figure. The figures below depict the same hyperbola as above, but shifted 2 units in the horizontal direction and 1 unit in the vertical direction. The construction is very similar to the preceding, except that the lines x = +- 4 are shifted 2 units to the right, becoming x = -2 and x = 6, while the vertical lines y = +- 5 are shifted to y = -4 and y = 6. The center is shifted from (0, 0) to (2, 1). The original hyperbola had equation y^2 / 25 - x^2 / 16 = 1. The shift is accomplished by replacing x by x - 2 and y by y - 1 to obtain the equation for the hyperbola: • equation of shifted hyperbola: (y - 1)^2 / 25 - (x - 2)^2 / 16 = 1. ________________________________________ Approach to asymptote: &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q013. The hyperbola you just sketched, which opens upward, has equation y^2 / b^2 - x^2 / a^2 = 1 and its foci lie at (0, c) and (0, -c). What is the equation of this hyperbola? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Y^2 / 25 - x^2 / 16 = 1 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThe foci of the parabola were given as (0, -c) = (0, -3) and (0, c) = (0, 3), so c = 3. The value of b is that of the y intercepts so b = 5. It follows that a^2 = b^2 - c^2 = 5^2 - 3^2 = 16, so that a = sqrt(16) = 4. The equation is therefore y^2 / b^2 - x^2 / a^2 = 1, or y^2 / 5^2 + x^2 / 4^2 = 1, or y^2 / 25 - x^2 / 16 = 1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q014. The hyperbola y^2 / b^2 - x^2 / a^2 = 1 has points on the 'top' and 'bottom', not on the 'right' and 'left' of the defining rectangle. How can you tell this from the equation? How can you tell that the equation x^2 / a^2 - y^2 / b^2 = 1 must have points at the 'right' and 'left' but not at the 'top' and 'bottom' of the defining rectangle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Because y^2 / b^2 - x^2 / a^2 = 1 has y intercepts at the b and -b points and x^2 / a^2 - y^2 / b^2 = 1 has x intercepts at the a and -a points. It depends on whether the x^2 is first or the y^2 is first. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aFor the equation y^2 / b^2 - x^2 / a^2 = 1, the y-intercept will lie at the x = 0 points, where y^2 / b^2 - 0^2 / a^2 = 1. Simplifying this equation we have y^2 / b^2 = 1, or y^2 = b^2 so that y = +-b. However if we attempt to find the x intercepts we substitute y = 0 and obtain -x^2 / a^2 = 1. This equation has no solution since x^2 and a^2 are both positive, making - x^2 / a^2 negative. Since the negative quantity cannot equal 1, there is no x intercept--so, there is no graph point on the right and left sides of the defining rectangle. For the equation x^2 / a^2 - y^2 / b^2 = 1 we see by letting y = 0 that the x intercepts are x = +-a, but if we let x = 0 we get the equation - y^2 / b^2 = 1, with no solution. We therefore have x intercepts but not y intercepts--i.e., points on the right and left sides of the rectangle but not on top and bottom. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q014. The hyperbola y^2 / b^2 - x^2 / a^2 = 1 has points on the 'top' and 'bottom', not on the 'right' and 'left' of the defining rectangle. How can you tell this from the equation? How can you tell that the equation x^2 / a^2 - y^2 / b^2 = 1 must have points at the 'right' and 'left' but not at the 'top' and 'bottom' of the defining rectangle? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Because y^2 / b^2 - x^2 / a^2 = 1 has y intercepts at the b and -b points and x^2 / a^2 - y^2 / b^2 = 1 has x intercepts at the a and -a points. It depends on whether the x^2 is first or the y^2 is first. confidence rating #$&*: 2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aFor the equation y^2 / b^2 - x^2 / a^2 = 1, the y-intercept will lie at the x = 0 points, where y^2 / b^2 - 0^2 / a^2 = 1. Simplifying this equation we have y^2 / b^2 = 1, or y^2 = b^2 so that y = +-b. However if we attempt to find the x intercepts we substitute y = 0 and obtain -x^2 / a^2 = 1. This equation has no solution since x^2 and a^2 are both positive, making - x^2 / a^2 negative. Since the negative quantity cannot equal 1, there is no x intercept--so, there is no graph point on the right and left sides of the defining rectangle. For the equation x^2 / a^2 - y^2 / b^2 = 1 we see by letting y = 0 that the x intercepts are x = +-a, but if we let x = 0 we get the equation - y^2 / b^2 = 1, with no solution. We therefore have x intercepts but not y intercepts--i.e., points on the right and left sides of the rectangle but not on top and bottom. " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!