2. Find the orthogonal trajectories of the ellipses 4x² +y? = 3x. 3. Determine the isogonal trajectories of the circles x? + y? = c if the angles of intersection are to be 60°

Q: Find the points on the ellipse x2+2y2=1, where f(x,y)=xy has its extreme values.

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Q: 1. Determine the orthogonal trajectories of the family of ellipses 16x² + 121y? = c. Sketch some of…

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Q: 14. Find the maximum and minimum values of the function f(x,y) y? = xy on the ellipse given by the…

A: Given f(x,y)=xy subject to    g(x,y)=x2+y24-1=0 We have to find the maximum and minimum values of…

Q: Draw the isoclines with their direction markers and sketch several solution curves, including the…

A: Given  dydx=y→       1             dyy=dx          ln  y=x+C            y=C1  ex     →        2…

Q: 5. Find the maximum and minimum values f(x, y) = xy can take on the ellipse 4x²+y? = 16.

A: The given function f(x,y)=xy to the ellipse 4x2+y2=16

Q: Find the equations of both the tangent lines to the ellipse x^2 +3y^2 = 9 that passes through the…

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Q: At what points does the curve r(t) = t i + (6t - t2)k intersect the paraboloid z = x2 + y2? (Enter…

A: Given: Equation of curve:                                    r(t)=t i +(6t-t2) k Equation of…

Q: Find the points on the ellipse 2x + 4y = 1 where f(x.y) = xy has its extreme values.

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Q: The surface of a mountain is modeled by the equation h(x, y) = 5000 − 0.001x2 − 0.004y2. A mountain…

A: Hints:  The given equation is h(x,y)=5000-0.001x2-0.004y2 and the given point is (500,500,3750). To…

Q: B) Use Lagrange multipliers to find the points on the ellipse 2x2+y? = 1, where f (r, y) = ry has…

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Q: Find equations of both the tangent lines to the ellipse  x2 + 4y2 =36 that pass through the point…

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Q: Suppose that the Celsius temperature at the point (x, y, z) on the sphere x² + y² + z² = 1 is…

A: Let's find lowest and highest temperature by using langrange multipliers.

Q: Show that line d:3x-4y-5=0 is tangent to hyperbola H:x2-4y2=5 and find the point of contact of the…

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Q: Find the point(s) on the hyperbola x^2 - y^2 = 2 closest to the point (0, 1).

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Q: Determine the orthogonal trajectories of the family of ellipses 16x² + 121y² = C. Sketch some of the…

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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: The orthogonal trajectories to the family of hyperbolas xy= C is given by the family 1)y=cx 2)y2 -x2…

A: We need to find the orthogonal trajectory to the family of hyperbola xy=c

Q: Find the length of the curve from the entire hypocycloid 2 2 x3 + Y³ = a³.

A: Explained below

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: for the given audratic equation determine, i- which way the parabolla opens ii- the cordinate of…

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Q: Approximate the length of the graph of the ellipse 4x2 + 9y2 = 36 in the first quadrant using…

A: Given equation of ellipse is 4x2+9y2=36. We have to find the length of the graph of the ellipse…

Q: At what point does the normal to y = 3x – (5 + 4x) at (1, –6) intersect the parabola a second time?…

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Q: Sketch the graph of the cylinder given by 4x2 + 9y2 = 36. show solution

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Q: An equation for the level surface passing through the point (2,0, - 1) is z =

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Q: Use the method of Lagrange multiplier to find the largest and least values of h(x, y) = x2y on the…

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Q: For each of the following quadratic equations in x and y, determine which conic section it is.…

A: Let Ax2+Bxy+Cy2+Dx+Ey+F=0 is a quadratic equation.Parabola: If B2-4AC =0 AND A=0 OR C=0Circle: If…

Q: Which of the following is the orthogonal trajectories of the family of ellipses 2x² + y² = a? xy = C…

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Q: The second order P.D.E =0 is © elliphic if xco,yco ® elliphic if xo © hyperbolic if x<o.geo ®…

A: The general form of a second-order partial differential equation is:…

Q: (a) Find the first order DE corresponding to the family of parabolas y = Cx² . (b) Find the equation…

A: a)The family of the curve of the parabola is y=Cx2 To find the first order differential equation…

Q: 1. Determine the orthogonal trajectories of the family of ellipses 16x + 121y = c. Sketch some of…

A: Given that equation of ellipses 16x2+121y2=cTo find equation of orthiginal trajectarySolution if…

Q: 3. he tangent to the ellipse I at the point P: V2 4. 3. 3w q(w) = 2w+ V2 %3D 3 q(w) = 3w V2w - V2 3…

A: Ellipse tangent

Q: 4. Find the two values of a for which the line connecting the point (1, –1) to a point (a, a²) on…

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Q: 4. Consider the equation of the hyperbola given by x2 + 2xy – y² = 7. Find an equation of the…

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Q: Find an equation of the parabola deseribed. Focus at (5, 0); vertex at (0, 0) O y = 20x2 O x = 20y O…

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Q: Ellipses

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Q: Find an equation of the ellipse with foci at (-5,9) and (-5,-10) and whose major axis has length 22.…

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Q: Find the maximum and minimum values of f(x,y)=xy on the ellipse  4x^2+y^2=6

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Q: 1. Determine the orthogonal trajectories of the family of ellipses 16x? + 121y² = c. Sketch some of…

A:

Q: 17. Let f(x, y) xy. Find the minimum and maximum values on the ellipse x² + 4y² = 4.

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Q: Use Lagrange multipliers to find the points on the ellipse x2+2y2=1, where f(x,y)=xy has its extreme…

A: This question is related to Lagrange multipliers theorem. We will solve this question by using…

Q: Evaluate x° ds where C is the part of the parabola y = x² from (0,0) to (2, 4).

A: Form the double integral and set the limits than integrate with respect to y and x.

Q: Find the minimum and maximum values of f(x, y) = x2y on the ellipse 4x2 + 9y2 = 36.

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Q: 1. Determine the orthogonal trajectories of the family of ellipses 16x² + 121y² = c. Sketch some of…

A:

Q: Draw the isoclines with their direction markers and sketch several solution curves, including the…

A: According to the given information, it is required to draw the isoclines with direction makers and…

Q: Find the maximum and minimum valves of f(xy)=5x+Y on the ellipse X+36y=!

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Q: Find the family of orthogonal trajectories of the family of hyperbolas x²-4y² =k

A: We need to find the family of orthogonal trajectories of the family of hyperbolas x²-4y² =k

Q: Let f(x, y) = x? + 4y. Find the extreme values of f on the ellipse x? + 2y? = 1.

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Q: 6. Show that if m is any real number, then there are exactly two lines of slope m that are tangent…

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Q: Find the family of oblique trajectories that intersect the family of the paraboles y =cx at angle e…

A: The answer is given below:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.