Advertisements
Advertisements
प्रश्न
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
Advertisements
उत्तर १
Let P(x, y) be a point on the curve y2 = 4x
which is nearest to the point A(2, 1).
Let D = AP
∴ D2 = (x – 2)2 + (y – 1)2
Put x = `y^2/4`
∴ D2 = `(y^2/4 - 2)^2 + (y - 1)^2`
Differentiating w.r.t. y,
`2D*(dD)/dy = 2(y^2/4 - 2)((2y)/4) + 2(y - 1)`
∴ `2D (dD)/dy = y(y^2/4 - 2) + 2y - 2`
`2D*(dD)/dy = y^3/4 - 2y + 2y - 2`
∴ `D*(dD)/dy = y^3/8 - 1`
`\implies (dD)/dy = 1/D(y^2/8 - 1)`
For extreme value of D, put
`(dD)/dy` = 0 ...(1)
∴ `1/D(y^3/8 - 1)` = 0
But D > 0
`\implies y^3/8 - 1` = 0
∴ y3 = 8
∴ y = 2
Now `D*(dD)/dy = y^3/8 - 1`
Again differentiating w.r.t. y,
`D(d^2D)/(dy^2) + ((dD)/dy)^2 = 3/8y^2`
But `(dD)/dy` = 0 ...[From (1)]
∴ `D*(d^2D)/(dy^2) = 3/8y^2`
∴ `(d^2D)/(dy^2) = 1/D*3/8y^2`
∴ `((d^2D)/dy^2)_(y = 2) = 1/D*3/8 xx 4`
= `3/(2D) > 0` ...(∵ D > 0)
∴ D is minimum when y = 2
∴ y2 = 4x gives x = 1
∴ The point (1, 2) is the point nearest to (2, 1).
उत्तर २
Suppose the required point on the curve is K(p, q) and the given point is A(2, 1).
∴ q2 = 4p ...(i)
Then, AK = `sqrt((p - 2)^2 + (q - 1)^2`
⇒ S = `sqrt((q^2/4 - 2)^2 + (q - 1)^2` ...[Let S = AK from (i) `p = q^2/4`]
Then, S2 = `(q^2/4 - 2)^2 + (q - 1)^2`
To find nearest point let us suppose S2 = T
Then, T = `(q^2/4 - 2)^2 + (q - 1)^2`
⇒ T' = `2(q^2/4 - 2) xx (2q)/4 + 2(q - 1)`
For critical points T' = 0
⇒ `2(q^2/4 - 2) xx (2q)/4 + 2(q - 1) = 0`
⇒ `(q^2/4 - 2q) + (2q - 2) = 0`
⇒ q3 = 8
⇒ q = 2 ...(ii)
To find the maxima or minima
T" = `(3q^2)/4 - 2 xx 2`
= `(3q^2)/4`
⇒ T" at 2 = `(3 xx 2 xx 2)/4`
= 3 > 0
Therefore, T is least
From (i) q2 = 4p
⇒ 22 = 4p
⇒ p = 1
Therefore, the required point is K(1, 2).
APPEARS IN
संबंधित प्रश्न
A telephone company in a town has 5000 subscribers on its list and collects fixed rent charges of Rs.3,000 per year from each subscriber. The company proposes to increase annual rent and it is believed that for every increase of one rupee in the rent, one subscriber will be discontinued. Find what increased annual rent will bring the maximum annual income to the company.
Find the maximum and minimum value, if any, of the following function given by f(x) = −(x − 1)2 + 10
Find the maximum and minimum value, if any, of the following function given by h(x) = sin(2x) + 5.
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:
`h(x) = sinx + cosx, 0 < x < pi/2`
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:
`g(x) = x/2 + 2/x, x > 0`
Prove that the following function do not have maxima or minima:
f(x) = ex
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
The point on the curve x2 = 2y which is nearest to the point (0, 5) is ______.
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
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.
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.
Choose the correct option from the given alternatives :
If f(x) = `(x^2 - 1)/(x^2 + 1)`, for every real x, then the minimum value of f is ______.
Solve the following:
A wire of length l is cut into two parts. One part is bent into a circle and the other into a square. Show that the sum of the areas of the circle and the square is the least, if the radius of the circle is half of the side of the square.
Solve the following:
Find the maximum and minimum values of the function f(x) = cos2x + sinx.
Find the local maximum and local minimum value of f(x) = x3 − 3x2 − 24x + 5
A rectangular sheet of paper has it area 24 sq. Meters. The margin at the top and the bottom are 75 cm each and the sides 50 cm each. What are the dimensions of the paper if the area of the printed space is maximum?
The function y = 1 + sin x is maximum, when x = ______
The two parts of 120 for which the sum of double of first and square of second part is minimum, are ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
If y = x3 + x2 + x + 1, then y ____________.
The distance of that point on y = x4 + 3x2 + 2x which is nearest to the line y = 2x - 1 is ____________.
Let f: R → R be a function defined by f(x) = (x – 3)n1(x – 5)n2, n1, n2 ∈ N. Then, which of the following is NOT true?
If the function y = `(ax + b)/((x - 4)(x - 1))` has an extremum at P(2, –1), then the values of a and b are ______.
The maximum distance from origin of a point on the curve x = `a sin t - b sin((at)/b)`, y = `a cos t - b cos((at)/b)`, both a, b > 0 is ______.
Determine the minimum value of the function.
f(x) = 2x3 – 21x2 + 36x – 20
If \[\mathrm{A}+\mathrm{B}=\frac{\pi}{2}\] then the maximum value of cosA.cosB is
