Advertisements
Advertisements
प्रश्न
Find the point on the straight line 2x+3y = 6, which is closest to the origin.
उत्तर
The equation of line is given as 2x+3y=6
`therefore y= (6 -2x)/3`
`therefore` The point of the line can be taken as P =`( x,(6-2x)/3)`
Co -ordinate of origin are O = (0,0)
Co-ordinates of origin are (0,0)= O
`OP = sqrt((x-0)^2 + ((6-2x)/3 -0))^2`
∴ OP = `sqrt(x^2 + (2- (2x)/3)^2`
∴ `OP^2 = x^2 + (4x^2)/9+4-(8x)/3`
`OP^2 = (13x^2)/9 - (8x)/3 + 4`
Let 2 ( ), OP f x OP is minimum when OP2 is minimum.
`therefore f(x) = (13x^2)/9 - (8x)/3 + 4` ....(1)
Differentiating w.r.t. ‘x’ we get,
`f'(x)= 26/9>0`
`therefore` OP is minimum at `x = 12/13`
`therefore y = 2-(2x)/3`
`therefore y = y - 2/3xx12/13 = 2- 8/13= (26-8)/13 `
`therefore y=18/13`
∴ The closest point on the line 2 3 6 x y with origin is `(12/13 18/13)`
APPEARS IN
संबंधित प्रश्न
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) = x3 − 6x2 + 9x + 15
Prove that the following function do not have maxima or minima:
h(x) = x3 + x2 + x + 1
A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is the maximum possible?
Show that a cylinder of a given volume, which is open at the top, has minimum total surface area when its height is equal to the radius of its base.
The volume of a closed rectangular metal box with a square base is 4096 cm3. The cost of polishing the outer surface of the box is Rs. 4 per cm2. Find the dimensions of the box for the minimum cost of polishing it.
Find the maximum and minimum of the following functions : y = 5x3 + 2x2 – 3x.
Divide the number 20 into two parts such that sum of their squares is minimum.
An open cylindrical tank whose base is a circle is to be constructed of metal sheet so as to contain a volume of `pia^3`cu cm of water. Find the dimensions so that the quantity of the metal sheet required is minimum.
Show that among rectangles of given area, the square has least perimeter.
Solve the following:
A rectangular sheet of paper of fixed perimeter with the sides having their lengths in the ratio 8 : 15 converted into an open rectangular box by folding after removing the squares of equal area from all corners. If the total area of the removed squares is 100, the resulting box has maximum volume. Find the lengths of the rectangular sheet of paper.
Determine the maximum and minimum value of the following function.
f(x) = `x^2 + 16/x`
Divide the number 20 into two parts such that their product is maximum.
A metal wire of 36cm long is bent to form a rectangle. Find it's dimensions when it's area is maximum.
If f(x) = x.log.x then its maximum value is ______.
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Show that the function f(x) = 4x3 – 18x2 + 27x – 7 has neither maxima nor minima.
If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of the triangle is maximum when the angle between them is `pi/3`
The maximum value of sin x . cos x is ______.
The maximum value of `(1/x)^x` is ______.
The maximum value of `["x"("x" − 1) + 1]^(1/3)`, 0 ≤ x ≤ 1 is:
Range of projectile will be maximum when angle of projectile is
The lateral edge of a regular rectangular pyramid is 'a' cm long. The lateral edge makes an angle a. with the plane of the base. The value of a for which the volume of the pyramid is greatest, is ______.
The maximum value of z = 6x + 8y subject to constraints 2x + y ≤ 30, x + 2y ≤ 24 and x ≥ 0, y ≥ 0 is ______.
A rod AB of length 16 cm. rests between the wall AD and a smooth peg, 1 cm from the wall and makes an angle θ with the horizontal. The value of θ for which the height of G, the midpoint of the rod above the peg is minimum, is ______.
If f(x) = `1/(4x^2 + 2x + 1); x ∈ R`, then find the maximum value of f(x).
Find the maximum profit that a company can make, if the profit function is given by P(x) = 72 + 42x – x2, where x is the number of units and P is the profit in rupees.
Find the maximum and the minimum values of the function f(x) = x2ex.
