2. Show that ( 1, 6), (0, 3), (1, 2), and (2, 3) lie on a common parabola.Hint: For scratch work, plug three of these points into y= ax2+ bx+c toobtain three linear equations determining a , b, and c.

Q: Find a parametric representation of the solution set of the linear equation.3x − 1/2y = 9

A: Given equation is Where x and y are unknown.

Q: Determine the general equation of a parabola opening downwards whose vertex is at (3,-4) with a…

A:

Q: For Exercise, find an equation of the form  y = ax2 + bx + c that defines the parabola through the…

A: Consider the given equation of parabola:

Q: dr if F= and C is a piece of the parabola y=4-x² from the point (–1,3) to the point (2,0).

A:

Q: Of all the points on the graph of z = 10 - x^2 - y^2 that lie above the plane x + 2y + 3z = 0, what…

A:

Q: Graph the quadric surfaces given by the equations in Exer- cises 41–48. y2 41. x² = y? x2 42. z2 9.…

A: As per guidelines incase of multiple questions (or) multiple sub-parts asked we are supposed to…

Q: If Ax+By+5z=C is an equation for the plane containing the point (0,0,1) and the line x-1=2, z=-63,…

A:

Q: give a geometric description of the set of points inspace whose coordinates satisfy the given pairs…

A:

Q: A nerd guitarist attempted to measure the auxiliary rectangle from the dips of his preloved electric…

A: The standard form of a hyperbola with the traverse axis on the x axis is x-h2a2-y-k2b2=1.  Here h,k…

Q: Consider the following equations of quadric surfaces: z =+-4 y? b) 4x² + (2) - - 2z? = -1 %3D | 16

A: The quadratic surfaces are given below-

Q: In the xy-plane, a line that has the equation y = c for some constant c intersects a parabola at…

A: in the xy-plane a line that has the equation y=c for some constant c intersects a parabola at…

Q: What is the equation of the parabola containing the points (0, 5), (–1,9), and (2, 3)? O y = r-z+5 O…

A: The standard form of a parabola is y=ax2+bx+c. The required parabola passes through the 3 points are…

Q: prove that the system of parabolas y2=4a(x+a) is self orthogonal

A: We show that the family contain both members with slope m and slope -1/m. Hence the family will be…

Q: Consider the problem of determining, in an Oxy orthogonal Cartesian system, the equation of the…

A: The objective is to determine the value of m for which the line will be tangent to the circumference…

Q: Find the quadratic surface and sketch the graph of r+y +- 4xyz = 1

A: The given equation is: x2+y2+z2-4xyz=1

Q: Find the vertex, force, and directrix of the parabola x^2=7y

A:

Q: If Ax+By+5z=C is an equation for the plane containing the point (0,0,1) and the line x-1=, z=-23,…

A:

Q: Find an equation of the largest sphere that passes through the point (-1, 1,4) and is such that each…

A: Find the equation of sphere.

Q: Any point on the parabola can be labeled (x, y), as shown. 4 3 2 (*, y) ¢ -6 -5 -4 -3 2 - 1 2 3 4 5…

A: According to figure equation of parabola in standered form X2=4bYnew equation of…

Q: Find the equation for the plane through the points Po (0, – 4, – 5), Qo(4,2,3), and Ro (0,3, – 2).

A:

Q: If 1x, y2 are the coordinates of a point P in the xy-plane, then x is called the _________  of P and…

A: If (x,y) are the coordinates of a point P in the xy-plane, then x is called the _________  of P and…

Q: Show, by completing the square, that the parabolay = ax2 + bx +c can be obtained from y = ax 2 by a…

A: The given equation of parabola is y=ax2+bx+c. Rewrite the above equation as follows.…

Q: Find b and c, given that the parabola y = x2 + bx + c passes through (0, 5) and (3, 23).

A: (0,5) means when x=0 then y=5. similarly, (3,23) means when x=3 , then y=23.

Q: 7. Show that the equation of the plane that has x-intercept a, y-intercept b, and z-intercept c…

A:

Q: Find the cartesian equation of the through (1, 3,-2) perpendicular to 51-71+3k. 5x-7y +3z - 22 0…

A: The required equation of the plane is : 5x - 7y + 3z + 22 = 0, which is the last option. The…

Q: The plane containing the point (2,5,0) and parallel to the plane 3x – 7y + 6z = 8 has equation Ах +…

A: Recall:      The equation of the plane parallel to Ax+By+Cz=D and passes through the point  α,β,γ ,…

Q: Consider a circle C in the xy-plane. If it is given that the points (2, 3),(2, −1) and (0, 3) are on…

A: Given points are (2, 3),(2, −1) and (0, 3)

Q: prove that the system of parabolas y² = 4a(x + a) is self orthogonal

A:

Q: (50p.) l- Find an equation of the plane through (1,3,–2) and the line of intersection of the two…

A: Given that, P1 : x-y+z=1 P2 : x+y-z=1 and Point A(1,3,-2)   We have to find Equation of Plane…

Q: Which of the following equations of parabolas has its vertex at (1, 2)? O y= 12 + 2x + 3 O y= r² +…

A:

Q: 7. If 1x, y2 are the coordinates of a point P in the xy-plane,then x is called the__________________…

A: Given that, x,y are the coordinates of a point P in the xy-plane. The objective is to fill in the…

Q: give a geometric description of the set of points inspace whose coordinates satisfy the given pairs…

A: Given that: y2+z2=1, x=0

Q: 1. Find the equation of the plane that passes through the points P(2,0,1), Q(1,0,-1) and R(3,0,1).

A: We have, Vectors in the plane are, Therefore, equation of plane passes through 3 points will be,…

Q: Show that (−1, 6), (0, 3), (1, 2), and (2, 3) lie on a common parabola.Hint: For scratch work, plug…

A: Consider the three points (−1, 6), (0, 3) and (1, 2). Substitute the above three points in the…

Q: Determine the points of intersection of the parabola y=2x^2 +12x +10 and the line y=4x+2

A: I am attaching image so that you understand each and every step.

Q: Find the canonical or standard equation of the sphere, if it exists. 2x² +2y² + 2z² – 4x+12y–16z+6=0…

A: To write the canonical or standard equation of the below sphere. 2x2+2y2+2z2−4x+12y−16z+6=0

Q: If Ax+By+5z=C is an equation for the plane containing the point (0,0,1) and the line X-1=と+2 3…

A:

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: x ds, where C is the portion of the parabola from (0,0) to (1, 1), followed by the line segment from…

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Q: What points in the xy-plane satisfy the equations and inequalities in Exercises 11–12? Draw a figure…

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Q: Show the complete solutions. Family of horizontal parabolas whose h to k ratio is 2. Ans: 2xy" -…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: =r such that the point (2, 1) is on the segment AB and the midpoint of Determine the coordinates of…

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Q: Find b and c so that y = 7x2 + bx + c has vertex (– 6, 2). - С —

A:

Q: What points in the xy-plane satisfy the equations and inequalities in Exercises 11–12? Draw a figure…

A: Since you have submitted two questions, we\'ll answer the first question. For the second question…

Q: For the three points ( 1,5), (2,9) and (-1, 7), find a quadratic model y = ax + bx + c through the…

A:

Q: Write an equation for the plane that contains the x-axis and the point (0, 5, 15). O 3 = x² + y² +…

A:

Q: 8. Represent the following inequalities in the Cartesian plane. a) 증+치 y² 9. 0 - y 1+ 1+ 1+ 7.3…

A:

Q: Given are two points [1,3] and (6, –1] in the plane. Which equation of the form a ·z + b•y+c=0…

A: (y-y1)/(y2 -y1) = (x-x1)/(x2-x1) (x1,y1 ) = (1,3) (x2,y2 ) =(6,-1) (y-3)/(-1-3) = (x-1)/(6-1)…

Q: 6.) Consider the points P(-1,2,5), Q(2, –3, 0) and (i) Write an equation of the plane containing t

A: Equation of plane: We know that, the equation of plane that passes through point (x, y, z) is given…

Q: Find an equation of the plane through the three points given: P=(0,0,3), Q=(2,−3,2), R=(−1,1,0)

A: find the equation of the plane

Maximization

Have you ever seen a bridge in the shape of a parabola? Where is its maximum on the bridge? The maximum of that bridge is the highest point on that bridge. The process of finding the maximum of the bridge here is called the maximization.