A rectangle is placed inside a semicircle of radius r, with one side along the diameter. What is the largest area the rectangle can have? There are at least two ways to set up variables you can use the coordinates of the upper corner of the rectangle, or you could use the angle the arrow makes with the diameter. Here, r is a 'parameter', to be treated as a fixed constant.

Q: 4. Find the equations of the line tangent to the intersection of the surfaces z = x´ + 2y and

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Q: Find the point(s) on x=3-y² that are closest the point A=(1,0)

A: Our Aim is to find the points on x=3-y2 that are closest the point A=(1,0).

Q: Find an equation of the tangent plane to the surface at the given point. x² + y² – 9z2 = 4, (-2, -3,…

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Q: Find the length of the curve 36y2 = (x — 4)3 , 2 ≤ y ≤ 3, y ≥ 0.

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Q: Find equations of both lines through the point (2, −30) that are tangent to the parabola y = x^2 -…

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Q: Find the distance from point P(1,1,1) to the pl r +y +z = 1

A: We have to solve

Q: An angle with its vertex at the origin of an xy-coordinate plane and with initial side on the…

A: We know, An angle with its vertex at the origin of an x-y-coordinate plane and with initial side on…

Q: Find the minimum distance from the point (8, 3, 4) to the plane x - y + z = 4. (Hint: To simplify…

A: Let ax+by+cz+d=0 is a plane  Then distance from the  a point (xo,yo,zo) is D= |axo +byo +czo…

Q: Find the point Q that is on the curve nearest %3D the point (4, 1)

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Q: Find the length of the curve parameterized by x = 2 + 3t² and y = 3 + 2t³ from t=0 tot = 3.

A: A detailed solution is given below

Q: Solve the center qnd radius of the circle passing through the points P(1,1) and is tangent to the…

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Q: Find the point(s) on x = 3 - y that are closest to point A = (1,0).

A: Given: Curve x=3-y2 and a point A(1,0).

Q: Find the parabola with equation y = ax2 + bx whose tangent line at (1, –1) has equation y = 3x – 4.…

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Q: 13. Show that the length of the portion of any tangent line to the ball x2/3 + y2/3 cut off by the…

A: Given: The equation of the ball is: x23+y23=a23...........(1) We have to show that the length of the…

Q: Eliminate the parameter from x = 2sint, y=4cost

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Q: What is the shortest distance from the surface ry + 15x + z2 = 204 to the origin? distance =…

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Q: 1. Point K is located at (4. 2, 0. 3)and point Lis located at (5. 8, -0.6). What is the location of…

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Q: Find the point on the line y = x closest to the point (1, 0). Hint: It is equivalent and easier to…

A: Distance between two points in xy-plane can be calculated using the distance formula. For two points…

Q: The first car A moves in the northern direction at a speed u(mph), and the second car B moves to the…

A: The first car A moves in northern direction i.e. towards positive y-axis and the second car B moves…

Q: Find the exact lengthy curve x=y4/4 + 1/8y2, where 1 is less than or equal too y is less then or…

A: The formula is L = ∫ab 1 + dxdy2dy

Q: Find equations for the lines tangent and normal to the cissoid of Diocles y (4 – x) = 9x° at (2,6).

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Q: man at point A on the shore of a circular lake of radius 1 km wants to reach point B on the shore…

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Q: Find the length of the curve y=x^3/2 from the point (1,1) to the point (4,8).

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Q: Find the point on the line y = 3x + 5 that is closest to the origim %3D (x, y) = %3D

A: We have to solve

Q: and the line perpendicular to Find the area of a triangle bounded by the y axis, the tine f(z) - 9 -…

A: We have to find the area of the region bounded by the y-axis, the line f(x)=9-67x, and the line…

Q: The projectile is thrown at the angle of 45 degrees with the initial velocity of 1 meter per second.…

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Q: Find the equation of the sphere of radius at centered at (1, 4, 5)

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Q: Find the shortest distance from the point (2,0, –3) to the plane x + y + z = 1.

A: This is a problem related to geometry part. Based on the concept of plane and point we will find the…

Q: Determine the center and the radius: X2 + Y2 + 2X +10Y + 1 = 0 O (-1,-5) Radius 5 O (1,5) Radius 5 O…

A: Topic-circle

Q: 4. Find the radius and center of curvature of the curve y=6x-x³ at the point (1,5).

A: This question is based on radius and centre of curvature formula.

Q: Show that the square of the length of subtangent at any point on the curve by 2 = (x+a)3,(b is not…

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Q: Evaluate the length of the curve x = 6 – Vy – 2 , 3< y<6

A: To find the length of following curve  x = 6 - sqrt( y - 6) 3 ≤ y ≤ 6

Q: At 6:00 am, ship A is 200 km west of port, heading east at 25 km/h. At the same time, ship B is…

A: Given:  At 6:00 am, ship A is 200 km west of Port, heading east at 25 km/hr.and at the same time,…

Q: Find the point on the parabola y = x^2 closest to the point (−3, 0)? and give the minimal distance?

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Q: The equation of a curved road is given as y = 100+6x–0.015x² +0.0000111? Determine the places where…

A: The equation of the curved road given is: y=100+6x-0.015x2+0.00001111x3 Measure of the "Sharpness of…

Q: A coast artillery gun can fire at any angle of elevation between 0° and 90° in a fixed vertical…

A: A coast artillery gun can fire at any angle of elevation between 0° and 90.

Q: A point moves such that the ratio of its distances from two fixed point is constant, show that its…

A: Let the moving point be P (x, y, z). Let A(a, 0, 0) and B(-a, 0, 0) be two fixed points.

Q: If 2x^4 + x^2y^2 - y^8 = 3x^2 + 4,what is the value of dyldx at point (4,1)? %3D

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Q: Atomic particles with like charges tend to repel one another. Suppose that two beams of like-charged…

A: Here, we can define the path of the given  particle as:

Q: How close does the semicircle y = V16 – x² come to the point (1, v5)? |

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Q: Find the least radius of curvature of the cubic parabola y=x3.

A: Given To find the radius of curvature, first find the curvature (K) using the formula. Find the…

Q: Find the orthogonal trajectories of the family of parabola described by the equation (y – 3) = = cx…

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Q: 3. Find the shortest distance from the origin to the plane x – 2y – 2z = 3. |-

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

Q: Atomic particles with like charges tend to repel one another. Suppose that two beams of like-charged…

A: x^2 − 4y^2 = 36 is a hyperbola Write it in the standard form and find a and b by comparing.

Q: Find equations of both tangent lines to the graph of the ellipse x2/4 + y2/9 = 1 that pass through…

A: We have given equation of curve x24+y29=1 and point 4,0 We will find slope of tangent at any point…

Q: 2. Circles with fixed radius r and tangent to the y-axis.

A: Solution is given below:

Q: Find the orthogonal trajectories of the family of parabola described by the equation (x+2} = cy.…

A: Explanation of the answer is as follows

Q: The figure shows a semicircle with radius 1, horizontal diameter PQ, and tangent lines at P and Q.…

A: As per the question we are given a circle of radius 1 and a horizontal line above the x-axis. Now we…

Q: Find the parabola y = ax2 + bx + c that passes through (2,1) and be tangent to the line y = 2x + 4…

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.