Let F= (y-r, z²+y) and C be the closed, piece-wise smooth curve in the zy-plane consisting of: C: the directed line segment from (a,0) → (a, a), followed by C: the directed line segment from (a, a) → (0, a), followed by; C3: the path along the quarter circle + y² = a? from (0, a) -→ (a,0). a) For what values of a >0 is fF. dr = 0? b) Is the line integral f.F- dr path-independent on R?? Explain.

Q: Let C be the positively oriented closed plane curve consisting of the line segment from (-3,0) to…

A:

Q: Determine the intersection if any between the line L₁: [x, y, z] = [3, 2, − 5] + r[0, 1, 4] and the…

A:

Q: Let R denote the rectangle with corners (−1, −2), (4, −2), (4,1), and (-1,1), and let P denote the…

A:

Q: Show that the following data can be modeled by a quadratic function. x 012 3 4 P(x) 6 8 12 18 26…

A: We will first complete the given blank table then from that we will check whether the second order…

Q: The vertices of AJKL are J(-3, 4), K(3, 0), and L(-5, -4). The equation of the median JLis y = 0.…

A:

Q: Consider the line which passes through the point P(5, -5, 5), and which is parallel to the line x =…

A: The direction vector of the line  given by:        x=1+4ty=2+1tz=3+3t is :    r→=123+t413 So, The…

Q: Let line l : x=y/2=z/3 and point P(1,0,1). Find an equation of the plane (in point-normal form)…

A: Solution of given question is written in below steps.

Q: Sketch the locus of points P(x, y) and write down its Cartesian equation given that P is equidistant…

A:

Q: (10) Find the distance of (3,-1,2) from each of the following: a) line: x = 1+ 2t, y = -t, z = 1-t…

A:

Q: 2) Find the point of intersection of the plane x - 5y + 3z = -127 and the line r(t) = (-2,-8, 7)t +…

A:

Q: Let L, be the line through the points (1,2,6) and (2,4,8). And let L2 be the line of intersection of…

A:

Q: x(t) Find a plane containing the point (6,-4,4) and the line y(t) z(t) 3 + 3t -3+ t 6 + 3t

A:

Q: 1) Find the distance from the point (1, 2, 3) (a) XZ plane to the (b) Z axis

A: In this question we will use basic of geometry i.e concept of 3D geometry. We will first plot the…

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.