ii) Prove that the straight line y = x is the curve giving the shortest distance between the points (0,0) and (1,1).

Q: What is the maximum vertical distance between the line y = x + 30 and the parabola y = x2 for -5 s x…

A: given, equation of line is y = x + 30 and equation of parabola is  we have to find maximum vertical…

Q: What is the shortest distance from the curve x^2+xy+y^2=16 to the origin?

A:

Q: Find the exact length of the curve. y = 3 + 2x^(3/2), 0 less than x less than 1

A:

Q: A line is tangent to the curve y = 3x1 and is bounded by the r- and y-axis in the first quadrant.…

A: Let's find.

Q: 5. What is the maximum vertical distance between the line y = x + 2 and the parabola y = x² for – 1…

A: The given equations are, y = x + 2       ....(1)  y = x2            ....(2) To find the maximum…

Q: Find the minimum distance from the point (2, -1, 1) to the plane x + y - z = 2.

A:

Q: 13) Find the points of intersecti on of the parabola and the line. y=2x^2-3x-1 and y=x+5

A: Equate the equations with each other and solve for x to find the point of intersection .

Q: Find the image of the line x = 1 under (a) flz) = z²

A:

Q: What is the maximum vertical distance between the line y = x + 51 and the parabola y= x² over the…

A: Given: The line y=x+51 and the parabola y=x2 To find: Maximum vertical distance between given line…

Q: Evaluate the length of the curve x = 5 - y+1,0<y<3

A: Given, x=5-y+1 ,     0≤y≤3

Q: A5: What is the minimum (perpendicular) distance from the point p = (-1,–2) to the line L := 4x = 14…

A:

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

A:

Q: Show that the curve y =(1 + x)/(1 + x2) has threepoints of inflection and they all lie on one…

A:

Q: What does it mean for two families of curves to be mutually orthogonal?

A: Two families of curve are said to be mutually orthogonal if they intersect at right angles. Angles…

Q: Find the exact length of the curve. 1 y = 4 -x² - In(x), 1< x < 8 2

A:

Q: Find all points on the graph of 3x 2 + 4 y 2 + 3xy = 24 where the tangent line is horizontal

A: Given:              3x2+4y2+3xy=24 We know the tangent line is horizontal when y'=0 So we want to…

Q: Find the minimum distance from the parabola x – y = 0 to the point (0, 3). Minimum distance = %3!

A: At P(x,y), the point of shortest distance the slope of the line from P to (0,3) must be…

Q: What is the shortest distance from the point (4,0, 5) to the sphere (x – 1)2 + (y + 2)2 + z² = 36?…

A:

Q: Determine shortest distance of the line 2x-3y-8 to the vertex of the curve x2 - 4a 4y +4 0. O 2 O…

A: We have given a line ,and find the shortest distance from vertex of parabola to line.

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

A:

Q: 4)Solve for the family of curves of straight lines with y-intercepts thrice as their corresponding…

A:

Q: 5. What is the maximum vertical distance between the line v1+2 and the parabolh v for

A:

Q: Find the length of the curve y = In (1 – 2*) over the interval [-,].

A:

Q: Determine the length of the curve x = 3y2 -1 between y = 0 and y = 4 3/2

A: The given curve is x=3y32-1 We will find the arc length of the curve between y=0 and y=4

Q: 8.) Find the exact length of the curve y= over the interval (0.4]. 2 %3D -

A: Given,          The exact length of the curve y=x22 over the interval 0,4 .

Q: The shortest distance from the point (5, 10) to the curve x2 = 12y is: 5.127 6.445 4.331 3.474

A:

Q: Find the length of the curve y=x^3/2 from (1,1) to (2,2squareroot2)

A:

Q: 3) Calculate the length of the curve interval [-2₁2]. y = x²/2 on the

A:

Q: ) Prove that the Point P(!,-) is a minimum Point of the curve y =lnx

A: Any point (x,y) is minimum for a given curve if it satisfies below two condition, a) First…

Q: The base of a three-dimensional figure is bound by the line x = y² and the y-axis on the interval…

A:

Q: 2 Find the point onthe curve y=Jx that is. the point (4,0). minimum distance from a

A:

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: + 2 Find the length of the curve y = from the point where a = 1 to the point where x = 2.

A: The length s of a curve with the function y=f(x) for the interval of x from a to b is given by the…

Q: 2 Show that the line y = x +3x+m for all values of m. y= mx+4 will touch or intersect the curve

A:

Q: What is the maximum vertical distance between the line y = x + 2 and the parabola y = x2 for -1 <x<…

A:

Q: 3- Find the length of the curves in Exercises 1-4. X = (y3/2/3) - y¹/2 from y = 1 to y = 9

A:

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: The equation of the curve in the tx-plane which passes through the point (2,3), and has, at every…

A: In this question, slope of curve and point is given and we have to find out equation of cuve.

Q: Find the shortest distance from the point (4,2) to the curve x^2 Зу^2 %3D 12 +

A:

Q: i) Find the exact length of the curve x5 1 15 4x3 y= + 1 < x < 2,

A:

Q: Show that the curves 2x2 + 3y2 = 5 and y2 = x3 are orthogonal.

A: To show that the two curves are orthogonal.

Q: Find the exact length of the curve. 23/2 12, 0 <x < 1 y =

A: Given : y=23x32,       0⩽x⩽1   To find : Curve length

Q: Find the exact length of the curve. X=1/3sqrt(y)(y-3), 9 less than or equal to Y less than or equal…

A:

Q: Find the exact length of the curve. y = 4 글2-글 In(x), 1 < x < 4

A:

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…

A:

Q: What is the maximum vertical distance between the line y = x + 38 and the parabola y = interval [–…

A: Given line equation is y=x+38 Given parabola equation is y=x2 From the above two equations x2-x-38=0…

Q: Find the exact length of the curve. 36y² = (x²-4)³, 5≤x≤9, y 20

A:

Q: 4 The line y= -Lx+3 has a perpendicu lar which fangent to the graph of y= (x²-2) (2x+k) at x=-l,…

A:

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.