In some designs of eyeglasses, the surface is "aspheric," meaning that the contour varies slightly from spherical. An aspheric lens is often used to correct for spherical aberration-a distortion due to increased refraction of light rays when they strike the lens near its edge. Aspheric lenses are often designed with hyperbolic cross sections. aspheric lens (3,0)

Q: Find the area of the surface generated when the curve y [2,4] is rotated about the y-axis.

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Q: Refer to Figure 25 in Section 11.5. Prove that the product of theperpendicular distances F1R1 and…

A: Prove that the product of theperpendicular distances F1R1 and F2R2 from the foci to a tangent…

Q: Find the radius of curvature of a parabola y² - 4x = 0 at the point (4,4). 2

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Q: 3.3.23. Write down the equation of the normal of the ellipsoid at (2,1, –1/2).

A: 3.3.23  we have given equation of ellipsoid x28+y24+z21=1   to find the normal to the ellipsoid at…

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Q: Suppose that a cylindrical container of radius r and height L is filled with a liquid with volume V…

A: Consider the given:  Express volume in terms of “h”, “r” and “w”.…

Q: Find parametric equations for the normal line to the surface z=-7x2 -9y2   at the point   (2,1,-37)

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Q: find a parametrization for the curve. the left half of the parabola y = x^2 + 2x

A: First, we will find the symmetric line of the given parabola y=x2+2x. We know that the symmetric…

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Q: Find an equation of the tangent plane to the given surface at the specified point. 2. (-16, 4, -1)

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Q: Find the area of the resulting surface if an arc of a parabola y = x² from (1,1) to (2,4) is rotated…

A: Given

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Q: Find the orthogonal trajectories for the family of ellipses with centers at the origin and…

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Q: By differentiating implicitly, find the slope of the hyperboloid x +y? - z = 1 in the r-direction at…

A:

Q: Find parametric equations and a parameter interval for the motion of a particle starting at the…

A: Given,

Q: A reflecting telescope contains a mirror shaped like a paraboloid of revolution.If the mirror is 4…

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Q: Find a parametrization of the osculating circle for the parabola y = x2 when x = 1

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Q: 3. Find the line normal to the ellipse at X=-1 y= 2+4x-x3

A: Given, y=2+4x-x3 At x= -1 y=2+4-1--13y=2-4+1y=-1 Now point -1,-1

Q: 3. Calculate the centroid of the given illustration below. y-axis 7.5ft- 4ft 2ft 5ft- х-ахis 2ft

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Q: Where does the normal line to the parabola y= x- x2 at  the point (1, 0) intersect the parabola a…

A: Normal to a curve is a line which is perpendicular to the curve at its point of intersection with…

Q: Derive the orthogonal trajectories of the family of hyperbolas ??=?.

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Q: The plane x = 1 intersects the paraboloid z = x2 + y2 in a parabola. Find the slope of the tangent…

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A:

Q: 3. Find the area of the surface cut from the nose of the paraboloid x = 1-y²-z2 by the yz-plane. Do…

A: Surface area in yz-plane

Q: Radiation is focused on an unhealthy area in a patient’s body using a semielliptical reflector,…

A:

Q: A mirror for a reflecting telescope has the shape of a finite paraboloid of diameter 0.3 meters and…

A: Equation of a circular paraboloid  f=r24d here, f= focal length along the axis of symmetry from the…

Q: Radiation is focused to an unhealthy area in a patient’s body using a semielliptic reflector,…

A: The width of the reflector is 100 cm. The height of the reflector is 30 cm.

Q: Find an equation of a generating curve given the equation of its surface of revolution. 8x2 + y2 +…

A: It is required to find the equation of generating curve given the equation of its surface of…

Q: The arc of the parabola y = x? from (1, 1) to (2, 4) is rotated about the y-axis. Draw the surface…

A: Given: Equation of parabola : y= x2  Arc of parabola passes through (1,1) and (2,4).

Q: (a) Where does the normal line to the ellipse x2- x   y + y 2 = 3 at the point (1, -1) intersect the…

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Q: Exer. 35-36: Find equations of the tangent line and normal line to the hyperbola at the point P. 3…

A:

Q: find a parametrization for the curve. the lower half of the parabola x - 1 = y2

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Q: Find an equation for the tangent plane to the hyperboloid x2+y2−z2=1 at the point (3,1,3).

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Q: Find an equation of the tangent plane to the given surface at the specified poin 8x2 + y2 - 7y, (1,…

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Q: A parabola has an equation of y²=2x, compute the radius of curvature at point (2, 2)

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Q: Find a parametrization of the osculating circle for the parabola y= X sequar when x=1

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Q: Exer. 35-36: Find equations of the tangent line and normal line to the hyperbola at the point P. 35…

A: As per guideline we have to solve one question

Q: A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating…

A:

Q: Determine the equation of the normal line to the hyperbola x^2 - 4y^2 = 9 at (5,2)

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A: Introduction :

Q: Determine the equation of the normal line to the hyperbola x2 – 4y2 = 9 at (5, 2).

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Q: Find the surface of revolution of the curve y = 4 + 3x² that is rotated about the y- axis between 1…

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Q: Find the parametrization for the curve  the lower half of the curve x+3=y^2

A:

Q: Find the equation of the tangent plane to the hyperboloid in 2 sheets x2 - y2 - z2 = 4 at the point…

A: We have to find the equation of the tangent plane to the hyperboloid x2-y2-z2=4 at the point 3,2,4.

Q: Find the equation of the sphere if one of its diameters has endpoints (8, -2, 8) and (11, 4, 17)…

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