The goal of this exercise is to "solve" a cubic equation by intersecting conics, similarly to the procedure by Omar Khayyam that we discussed in class. (a) Show that if there exists a point P = (x, y), x > 0, such that P lies in the intersection of the hyperbola xy = d with the parabola y² + dx - db = 0, where d, b>0, then x will be a positive solution of the cubic equation x³ + d = bx². (b) Illustrate geometrically how this procedure can lead lead to zero, one or two positive solutions, depending on the relative positions of the hyperbola and the parabola.

Q: The parabola y = ² is changed to the form y = a (x − p)² + g by translating the parabola 2 units…

A: The parabola y=x2 is changed to the form y=ax-p2+q by translating the parabola 2 units down and 3…

Q: (b) Find an equation for the parabola that has axis x = -1 and passes [3] through the points (4,5)…

A: Directrix is a line which together with a point called the focus defines the set of points that form…

Q: Find the value of Y for the hyperbola defined by the equation y² – x² = 7 at the point (3, 4). dx2

A: The equation is y2-x2=7. First find the value of dydx as follows.…

Q: Two stations that are 2 km apart receive a sound signal issued from a source located at point…

A: The distance between the point 0,-1 and the point P (x, y) is, d1=x2+y+12 The distance between the…

Q: 1. A parabola has equation -12x + y? – 24 = 0. Write the equation in standard form.

A: The equation of the parabola is, - 12x + y² - 24 = 0 So, y² = 12x + 24

Q: Given hyperbola   (y + 1)2   −   (x − 2)2   = 1       9                4         what will its…

A: Given: An equation of the hyperbola, y+129-x-224=1.

Q: (y+1)? (x-2)2 2.Find the standard equation of the hyperbola whose standard equation is 9. %3D 16

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Q: (b) Find for what value of the equation 12x²+36xy + y² + 6x + 6y + 3 = 0 represents a pair of…

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Q: Find b and c so that the parabola with equation y = 4x2 - bx - c has a vertex at (2 , 4)?

A: Using the vertex formula we find the x value of the vertex[formula: x value of vertex of Ax^2+Bx+C=0…

Q: What is the equation of the line tangent to the hyperbola = 1 at the point (V2a, b)? y = -a+b 26 V26…

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Q: O Find the equation of the parabola that has its focus at (-4, 1) and its directrix x = 0.

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Q: (3) Find the coordinates of the points of intersection of the parabola y = x2 - ax and the line y =…

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Q: Solve for the directrix of the parabola y2 + 8x = 32

A: We have to solve for the directrix of the parabola y2 + 8x = 32.

Q: Find the values of integers,, and c that will make the hyperbola -9x² + 4y² + ax +by+c= 0 pass…

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Q: (z – #1)² (y – yn)² 1 to find the y-values to the nearest integer from the given z and y-values in…

A: We have to find out y value

Q: 2. Which equation represents a parabola with vertex (4,4) and directrix y = 2? Oy=2x² - 4x + 4…

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Q: Which is the equation of a hyperbola with co-vertices at (0, 7) and (0, −7) and a transverse axis…

A: Ans (1) we have givenCo=Vertics of hyperbola (0,7) and (0,-7)⇒Compairing with co-vertice from (0, ±6…