If the intersection points of the hyperbola 3x² – y² = 11 and the line 2x + y = 10 are P and Q, which of the following is the midpoint of [PQ]? A) (20,30) B) (20,-30) C) (-20,30) D) (18,-24) E) (18,-30)

Q: Give the general equation of the plane containing the point (3,-1,4) and perpendicular to the line =…

A:

Q: Given the focus and directrix of a parabola, identify its important characteristics. 10 8 -10 -8 -6…

A:

Q: find aplane through Po (2,!,-1) and perpendicular to the line of Intersectionof the planes 2xty-Z=3)

A:

Q: The solid whose base is the triangle with vertices (0, 0), (2, 0), and (0, 2), and whose cross…

A:

Q: *5. Prove that no two lines in hyperbolic geometry are equidistant from one another by showing that…

A: To prove that no two lines in hyperbolic geometry are equidistant from one another , first we…

Q: Calculate the distance from point A(1, 2, 1) to the plane of equation 3x-4y + 5z = 1.

A: To find distance of point to plane using the formulae: Given point is and plane equation is .

Q: Find an equation for the plane in R³ spanned by the points (1,5,

A:

Q: The plane passes through the point (18, -16, 9) and contains the line given by = = z . 1.

A:

Q: A parabola is the set of all points in the plane that are and a equidistant from a fixed point…

A: Given: A parabola is the set of all points in the plane that are equidistant from a fixed point…

Q: Find a parametrization for the line segment beginning at P,(4, - 7,- 7) and ending at P2(0, - 7,2).

A: The line segment beginning at P1<4, -7, -7> and ending at P2<0, -7, 2>

Q: A quadrilateral has coordinates R(-5,2),S(-1,3), T(-2,–1) and U(-6,– 2). Verify algebraically, that…

A: Let X be mid point of R(-5,2) and S(-1,3) ∴Coordinate of X is -5-12,2+32…

Q: If the line X equals three is the axis of symmetry of a para bola , Which coordinates below could be…

A: The axis of symmetry is a line, that divides the parabola into 2 equal halves. For the parabolas of…

Q: Let S be the quadric with equation 36x² – 4y² – 9z2 – 36x = 0 • Classify the graph of S…

A: Given, S is the quadric with the equation                                36x2-4y2-9z2-36x=0

Q: A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100 feet.The…

A: Given: A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100…

Q: 6. Determine the distance from the point (2, -1, 5) to the plane 3x - 3y + 5z = 8.

A:

Q: In a flat geographic region, the course of two rivers can be represented approximately by the…

A:

Q: Find the distance between the point (3,4,5) and the plane 2x + 4y 3z = 12.

A:

Q: Find the distance from the point Q(1, 2, -4) to the plane 2x - 5z = 5.

A:

Q: The end points of line segment 4, which lies in the xy-plane, are (-2, 5) and (6,-3). Which of the…

A: i have used the formula of midpoint of the two points and the slope point form of the straight line…

Q: Consider the point (3,4,5) What is the projection of the point on the xy-plane?

A: The given point is (3,4,5).

Q: opens upward with vertex at (-12,-6) with directrix y=- 17

A:

Q: Find an equation of the plane that passes through the point (1, 6, −4) and contains the line:

A:

Q: A triangle is inscribed in a rectangular hyperbola such that the tangent at one of the vertices is…

A:

Q: Find an equation of the sphere that passes through the point (1, 8, 5) and has center (3, 1, −3).

A: We have to find equation of sphere.

Q: Find the distance from point P(2, 7, -6) to the plane of equation 5x - y + 4z - 8 = 0

A: We have to find distance

Q: Find the equation of the plane con taining the points Р(4, -1, 3) Q(1, -2, -1) and R(1, 0, 8)

A:

Q: A hyperbola is the set of points in a plane, the difference of whose distances from two fixed points…

A: Given : A hyperbola is the set of points in a plane the difference of whose distances from two fixed…

Q: If the coordinates of the ends of a focal chord of the parabola y²=4ax and (x,,y,) and (x,.y;) then…

A: Consider the given parabola y2=4ax Suppose Ax1 , y1  and  Bx2 , y2 be the ends of the focal chord of…

Q: Find the distance from the point (-3, 4, -1) to the plane 5x−4y−5z=−3.

A:

Q: Find an equation of the sphere with center (2,-3,6) that touches the xy-plane.

A:

Q: Which of the points A(-4, 0, -1). B(3, 1, -1), and C(2, 4, 6) is closest to the yz-plane? Which…

A:

Q: Verify that parallelogram ABCD with vertices A(-5, – 1), B(-9, 6), C(-1, 5), and D(3, – 2) is a…

A: We can verify this as below using the diagonals.

Q: How many arrangements can be made using 2 letters of the word HYPERBOLAS if no letter is to be used…

A: To find the number of arrangements which can be made using two letters of the word HYPERBOLAS if…

Q: A parabola is the locus of a point equidistant from?

A: Answer : A parabola is the locus of a point equidistant from a fixed point and a fixed line.

Q: Find the distance from the point (1, -4, 4) to the plane −2x−4y+3z=5

A: Given:The point (1, –4,4) and the equation of the plane –2x–4y+3z=5.The distance from the point…

Q: The length of the major axis of the cllipse with center (-4, 1), minor axis parallel to the y-axis…

A: Equation of ellipse with center (-4,1) is given by

Q: What is the distance from the point (1,9,3) to the xz-plane?

A:

Q: What is the distance from the point S(-2, 3, -2) to the line of intersection of the planes 2x + 3y +…

A:

Q: Find an eguation of the sphere that passes through the point (5,-2,-1) whose whose Center is the…

A:

Q: Prove analytically that the midpoint of the hypotenuse of a right triangle is equidistant from the…

A:

Q: Explain why a line in hyperbolic geometry must contain two distinct ideal points.

A: We have to answer given Questions.

Q: The equation of the sphere with center (7,-10, 1) that touches that y-axis is

A: To find the equation of the sphere we need center of the sphere  radius of the sphere.  Here given…

Q: A parabola is the set of all points in the plane that areequidistant from a fixed point called the…

A:

Q: Which of the points A(−4, 0, −1), B(3, 1, −5), and C(2, 4, 6) is closest to the yz-plane? Which…

A: Given- A(−4, 0, −1), B(3, 1, −5), and C(2, 4, 6) To do- 1.Which of the points A(−4, 0, −1), B(3, 1,…

Q: Find the distance from the point (-2, 1, 4) to the plane y = -5

A: Given plane is: y = -5 Given point is: (-2, 1, 4) We know that the distance (d) of a point P(h, j,…

Q: find the orthogonal trajectory of family of parabolas represented by the equation…

A: To find orthogonal trajectories of  y2=4ax.

Q: Find the distance from the point (-4,3,-2) to the yz-plane.

A:

Q: what is the equation of the geodesic in the hyperbdic upper half-plane that joins -1+i with Iti

A:

Q: Find the orthogonal trajectory for the family of parabolas with center at the origin and focion on…

A:

Q: Find the distance from the point (-1,4,5) to the plane 3x+2y-4z=5?

A:

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.