Find the point on the parabola x = t, y = t2, -∞ < t < ∞ closest to the point (2, 1/2).

Q: At what point does the normal to y = 2 + 2x + 2x² at (1,6) intersect the parabola a second time?

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Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 ≤ t ≤ 2π, closest to the point (3/4, 0).…

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Q: The cubic y = ax3 + bx? + cx + d tangent to the line y = 12x + 13 at (-1, 1) and have a horizontal…

A: Solution...

Q: Find an equation of the plane tangent to the graph of g(r, y) = 3ev²-z² at the point where (x, y) =…

A:

Q: Find the maximum value of ƒ(x, y) = 6xye^(-(2x+3y) )in the closed first quadrant (includes the…

A: Given and

Q: The tangent of the parabola x = y² + y + 1 at the point (1,0) is

A:

Q: Find the equations of the tangent line in the given plane. Tangent to the parabola z = x² + 3y² , x…

A: Tangent line equation problem....

Q: Find all points on the ellipse x2 - xy + y2 = 1 at which the tangent line is horizontal.

A: The given equation of ellipse is,

Q: 8. At what point of the parabola is the tangent line parallel to 7x+y-2=0? y = 2r-3x +1 O (1,0) O…

A: Here is the solution:

Q: Find the asymptotes of the cissoid of Diocles defined by the equationy2(2a−x)=x3,a&gt;0.

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Q: Write the general solution of the fourth-order DE y(4) – 2y" + y = 0 entirely in terms of hyperbolic…

A: We have to find the general solution of the differential equation y4-2y''+y=0. The auxiliary…

Q: Find a point on the curve y = x² that is closest to the point (18, 0). %3D AY y =2 (x, y) (18, 0)

A: Find a point on the curve y=x2 that is closest to the point  Here use distance formula…

Q: Find . F dR such that F(x, y) = (2y, xy – 2x) And I is the curve in the 3rd quadrant from the point…

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Q: Find the point on the ellipse x = 2 cos t, y = sin t, 0 &lt;=t&lt;= 2pai closest to the point (3/4,…

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Q: 8. Calculate fo F • dr, where F(x, y) = y² i + (6x + 6y)j, and C is the portion of the parabola 2y²…

A: Given curve is 2y2-x=1 and points are (1,-1) to (7,2)

Q: (6) Find the points on the curve 5x2 + 6xy + 5y² = 9 which are nearest to and farthest from the…

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Q: ind the equations of the tangent line to the ellipse x² + 4y² + z² = 9, x = 2 at the point (2, -1,…

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Q: Find the tangent/sto the parabola (x + 2y)2 + 2x – y – 3 = 0, parallel to the line x – 4y = 2.

A: The equation of the parabola is given as  x+2y2+2x-y-3=0. On simplifying,…

Q: Find the tangent plane to the equation z = 3e²²-2y at the point (4, 8, 3)

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Q: Find an equation of the parabola y = ax2 + bx + c that passes through (0, 1) and is tangent to the…

A: Given" Parabola y = ax2 + bx + c that passes through (0, 1) and is tangent to the line y = x − 1 at…

Q: 3. Find an equation of the tangent plane to the paraboloid defined by z= 5x +4y? at the point…

A: The given equation is; z=5x2+4y2 at the point (-1,1,9).

Q: 7. For whit value of b will the line y =- y 3x +4x -1? 2x+ b be tangent to the parabola

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Q: Find the point on the curve y = Vx closest to the point (1,0).

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Q: Show that the equation of the tangent ine to 'the graph of the parabola at the point Pt'=(t,t²) 1s…

A: Consider the given function whose equation of tangent is required to calculate.

Q: Find the exact length of the curve y = In(1 – x²), 0 Sx S

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Q: Find the Maximum value of the function f(x, y) = x² – y² – 2y on the curve x? + y? = 25. %3D -

A: By langrange multiplier method n

Q: What is the maximum vertical distance between the line y = x + 42 and the parabola y = x for -6 s xs…

A: Consider the equation y=x+42 and y=x2; -6≤x≤7. Substitute y=x+42 in y=x2.…

Q: Find the points on the curve x2 + xy + y2 = 1 in the xy-plane that are nearest to and farthest from…

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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 the minimum distance from the parabola: x - y2 = 0 to the point (0,3).

A: x - y2 = 0 x = y2  Let us take a point (x,y)=(y2, y) on the parabola

Q: Find the length of the parabola given by c(t) = (t, t2) for 0 ≤ t ≤ 1.

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Q: 4. Determine the constant a,b, c, and d so that the curve defined by f(x) = ax³ + bx² + cx + d has a…

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Q: Q2/ Find the first moment about y-axis of the region in the first quadrant bounded by the x-axis,…

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Q: Find an equation for the level surface of the function f (x, y) = ln (x² + y²+ z²) that passes…

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Q: 4x2 dx - xy dy 16x2, y = o from (-1, 0, 16) to (1, 0, 16) (d) dz, where c is the parabola z

A: To evaluate the line integral along the arc C of the given parabola

Q: Find a parabola with equction Y=ax +bx+c that has a Slope I at x -1, slope 9 at x=-1, and passes…

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Q: Find the arclength of y = 4x2 - 4 where x is greater than or equal to 0 and less than or equal to 4

A: Concept Used: The arc length of the function is the length of the curve y=fx on the interval a,b is…

Q: The tangent and the normal to the parabola y²=20x at the point T(5,10) meet the x-axis at the points…

A: Details solution is given below:

Q: Find the equation of the tangent plane to the surface z = x² +y² + 1 at the point (1,1,3). Write the…

A: use the equation of tangent  plane for the given surface.

Q: Find an equation of the parabola y = ax² + bx + c that passes through (0, 1) and is tangent to the…

A:

Q: Find the equation of tangent line to the parabola y² = 6x – 3 which is perpendicular to the line x +…

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Q: What is the maximum vertical distance between the line y -x = 2 and the parabola y = x? in the…

A: We have to find the maximum vertical distance between the line y-x=2 and the parabola y=x2 in the…

Q: Find the tangent line to the ellipse x^2+4y^2=8 parallel to the line x+2y=6.

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Q: 2. Find the shortest distance from the point (2,0) to the curve: (a) y = 3x – 1 (b) y = x^2 (c) x^2…

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Q: Compute y ds over the part of the parabola y =2Px, for P > 0 starting at (0,0) at (2P,2P).

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Q: Find the equations of the tangent line in the given plane. Tangent to the parabola z = x² + 3y² , x…

A: Given :                    z = x2 + 3y2      at the point (1,2,13)

Q: (a) Prove that the line 2x 3y + 26 =0 is a tangent to the curve x + y2 - 4x + 6y – 104 = 0 hence…

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Q: Find an equation for the tangent plane to z = 9x? – 8y² at the point (1, 1, 1). - = 0.

A: Here we find the equation of tangent plane.

Q: find the under the curve y=7-t^2 and above the t-axis

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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.