Find an equation for the curve traced by the point P(x, y) if the distance from P to the vertex of the parabola x2 = 4y is twice the distance from P to the focus. Identify the curve.

Q: Eliminate the parameter to find a cartesian equation of the curve. x = et,  y = e-4t

A: A curve is a two dimensional object with equation of the form y=f(x). This equation can sometimes be…

Q: Find the exact length of the curve.   y= 2/3x3⁄2,    0 ≤ x ≤ 4

A: Given query is to find the length of the curve for the interval.

Q: Find the parabola with equation y=x^2+bx+c that has slope of 8 when x=9 and passes through the…

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Q: Find a parabola with equation y = ax2 + bx + c that has slope 8 at x = 1, slope −12 at x = −1, and…

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Q: find a cartesian equation for the curve r2 = 14

A: Find a cartesian equation for the curve         r2 = 14

Q: An ellipse has the equation 16x² + 25y = 40O. Determine the equation of the tangent of the curve at…

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Q: 4. Find the focus, directrix and vertex of the parabola (x + 2)2 = 3y - 6, then sketch the curve.

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Q: Find the standard equation of the parabola with x intercepts at x = 2 and x -3, and a y-intercept at…

A: Solution is given below

Q: Graph the given parabola and find its focus: Y=X squared

A: The given equation of parabola is: y=xory2=x  where x > 0

Q: Find a parabola with equation  y = ax2 + bx + c  that has slope  13  at x = 1, slope  −19  at x =…

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Q: Find the angle at which the parabola y=x2−14 intersects the positive x−axis.

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Q: Eliminate the parameter to find a Cartesian equation of the curve: x = VE ,y = 1 – t and then sketch…

A: Given,x=t...............(1)andy=1-t...................(2)differentiating both the equation w.r.t.…

Q: Consider the parabola y = -x +4x + 9 and the line y=2x + 1. Solve for the x-coordinates at their…

A: x - coordinates of the point of intersection = ?

Q: Find the equation of the asymptote with a positive slope for the following hyperbola. (x + 8)? 1 (y…

A: Given hyperbola s standard equation of hyperbola : Therefore,

Q: Find the equation of the parabola with vertex at the origin and focus F (0, 5)

A: General equation of parabola with vertex at origin and focus on y axis is x^2= 4ay.

Q: Find an equation for the indicated half of the parabola.Lower half of (y + 1)2 = x + 3

A: Given equation is y+12=x+3.

Q: FIND THE PROPERTIES OF THE CURVE AND SKETCH FULLY   Y2 = x2+5x−4/x−5

A: Given,           y2=x2+5x-4x-5

Q: Find an equation for the parabola that has its vertex at the origin and satisfies the given…

A: Given vertex is (0 , 0) and focus on negative y axis focal diameter is 2

Q: Find the coordinates of the point in the curve y = 12x - 2y + 23 that is nearest to the point…

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Q: Find an equation for the parabola that has its vertex at the origin and satisfies the given…

A: Given: Focus on the negative y-axis, 6 units away from the directrix

Q: Find equations for two lines through the origin that are tangent to the ellipse a- 8x +y + 3 =0.

A: We are asked to find the equation of two lines that are tangent to the ellipse.

Q: 3. Find the equation of the parabola with focus at F(3,2) and with the line x = -4 as directrix.

A: Topic- equation of parabola from focus ans directrix.

Q: Find a point on the parabola y - x* + 9 which is closest to the origin.

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Q: y =(+ + +)* x = 4t + 1, 4

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Q: Find the length of the curve. x- 4t°, y= 6t, 0sts 3

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Q: Find the coordinates of the point on the parabola y = x² which is closest to the point (3, 0).

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Q: find the points on the parabola y=x^2 closest to the point (0,1)

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Q: Find an equation of the parabola in standard form: with focus at F(7,-1) and directrix 6 units from…

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Q: Find the shorted distance from the point A(2, to a point on the parabola y = x². And find the point…

A: Let (x,y) = (x,x2) point on the parabola is closest to A Distance between (2 , 1/2) and (x,x2) is:…

Q: Find the equations of the osculating circles of the parabola y = 3(x – 2)2 at the points (2,0)

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Q: Find an equation for the parabola that has its vertex at the origin and satisfies the given…

A: Given- The parabola that has its vertex at the origin and satisfies the given condition(s).…

Q: Find an equation for the parabola shown in the figure.

A: We have given Vertex: 1, 0 Focus:6, 0   The given graph is symmetric about x-axis. Equation of the…

Q: Find the following points of intersection. The point(s) of intersection of the parabola y = x2 + 2…

A: Given equation of parabola is y=x2+2 and equation of line is y=x+4. We have to find point of…

Q: Find the point(s) on the parabola x =y closest to the point (0,3)

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Q: Find the point on the parabola * +y“ that is closest to the point (0, -3). =

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Q: FInd the equations of the tangents with slope m=-2/9 to the ellipse 4x^2+9y^2=40

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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: The equation of the curve is given as y = 3x + 1

A: topic - slope of curve at a point

Q: Find the point on the parabola y 2 =2x that is closest to the  point ( 1, 4)

A: Given: y2=2x let's point on parabola is t22,t now, if point1,4 is closest to t22,t2 then product of…

Q: Find the exact length of the curve. 1 y = + 10x37 6 1 ≤ x ≤ 3

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Q: Eliminate the parameter to find a Cartesian equation of the curve. x = t2,    y = t5

A: A curve is a two-dimensional object with equation of the form y=f(x). Curves can be parameterized to…

Q: Find equations for two lines through the origin that are tangent to the ellipse x - 8x + y² + 5 = 0.

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Q: Find the properties of the curve and sketch fully   y2=x^2-4x-12/x^2-2x-3

A: I have provided this answer in step 2.

Q: FIND THE PROPERTIES OF THE CURVE AND SKETCH FULLY 6. y2 = x²+5x-4 x-5

A: Given, the function y2=x2+5x-4x-5. We have to sketch the given curve. Also, find the properties.

Q: Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C,…

A: The given parametric curve is, x=2ty=8t3-1≤t≤1 From the parameter of x, x=2tt=x2

Q: Find the point P on the parabola y = x2 closest to the point (57, 0). (х, у) %3D ]-«

A: Given the equation of parabolay=x2         ........(1)

Q: Find the parabola with equation y = ax? + bx whose tangent line at (2, 8), is y = -2x + 12.

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Q: find a cartesian equation for the curve r=3cos(θ)

A: Cartesian equation for the curve is r=3 cos(θ)   Now x=r cos(θ)  …….. equation(1) cosθ=xrr=3xcosθ…

Q: Find an equation of the parabola that satisfies the given conditions.Focus F(0, -4), directrix y = 4

A: To find: The equation of the parabola that satisfies the given conditions. Given information: Focus…