Advertisements
Advertisements
प्रश्न
Find the point on the curve y2 = 4x which is nearest to the point (2,\[-\] 8).
Advertisements
उत्तर
\[\text {Let point }\left( x, y \right)\text { be the nearest to the point } \left( 2, - 8 \right) . \text { Then }, \]
\[ y^2 = 4x\]
\[ \Rightarrow x = \frac{y^2}{4} . . . \left( 1 \right)\]
\[ d^2 = \left( x - 2 \right)^2 + \left( y + 8 \right)^2 ............\left[ \text {Using distance formula } \right]\]
\[\text { Now,} \]
\[Z = d^2 = \left( x - 2 \right)^2 + \left( y + 8 \right)^2 \]
\[ \Rightarrow Z = \left( \frac{y^2}{4} - 2 \right)^2 + \left( y + 8 \right)^2 ................\left[ \text { From eq. }\left( 1 \right) \right]\]
\[ \Rightarrow Z = \frac{y^4}{16} + 4 - y^2 + y^2 + 64 + 16y\]
\[ \Rightarrow \frac{dZ}{dy} = \frac{4 y^3}{16} + 16\]
\[\text { For maximum or minimum values of Z, we must have }\]
\[\frac{dZ}{dy} = 0\]
\[ \Rightarrow \frac{4 y^3}{16} + 16 = 0\]
\[ \Rightarrow \frac{4 y^3}{16} = - 16\]
\[ \Rightarrow y^3 = - 64\]
\[ \Rightarrow y = - 4\]
\[\text { Substituting the value of x in eq. } \left( 1 \right), \text { we get }\]
\[x = 4\]
\[\text { Now, }\]
\[\frac{d^2 Z}{d y^2} = \frac{12 y^2}{16}\]
\[ \Rightarrow \frac{d^2 Z}{d y^2} = 12 > 0\]
\[\text { So, the nearest point is }\left( 4, - 4 \right) .\]
APPEARS IN
संबंधित प्रश्न
f(x)=| x+2 | on R .
f(x) = | sin 4x+3 | on R ?
f (x) = \[-\] | x + 1 | + 3 on R .
f(x) = 16x2 \[-\] 16x + 28 on R ?
f(x) = (x \[-\] 5)4.
`f(x)=2sinx-x, -pi/2<=x<=pi/2`
f(x) =\[x\sqrt{1 - x} , x > 0\].
f(x) =\[\frac{x}{2} + \frac{2}{x} , x > 0\] .
`f(x)=xsqrt(32-x^2), -5<=x<=5` .
If f(x) = x3 + ax2 + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?
f(x) = (x \[-\] 1)2 + 3 in [ \[-\] 3,1] ?
f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in }[1, 9]\] .
Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 − 15x2 + 36x + 1 on the interval [1, 5].
Divide 64 into two parts such that the sum of the cubes of two parts is minimum.
Divide 15 into two parts such that the square of one multiplied with the cube of the other is minimum.
A wire of length 20 m is to be cut into two pieces. One of the pieces will be bent into shape of a square and the other into shape of an equilateral triangle. Where the we should be cut so that the sum of the areas of the square and triangle is minimum?
Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.
A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?
Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]
An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .
Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r.
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?
Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?
Find the maximum slope of the curve y = \[- x^3 + 3 x^2 + 2x - 27 .\]
The total cost of producing x radio sets per day is Rs \[\left( \frac{x^2}{4} + 35x + 25 \right)\] and the price per set at which they may be sold is Rs. \[\left( 50 - \frac{x}{2} \right) .\] Find the daily output to maximum the total profit.
Manufacturer can sell x items at a price of rupees \[\left( 5 - \frac{x}{100} \right)\] each. The cost price is Rs \[\left( \frac{x}{5} + 500 \right) .\] Find the number of items he should sell to earn maximum profit.
The strength of a beam varies as the product of its breadth and square of its depth. Find the dimensions of the strongest beam which can be cut from a circular log of radius a ?
Write necessary condition for a point x = c to be an extreme point of the function f(x).
Write the maximum value of f(x) = x1/x.
For the function f(x) = \[x + \frac{1}{x}\]
The function f(x) = \[\sum^5_{r = 1}\] (x \[-\] r)2 assumes minimum value at x = ______________ .
The least value of the function f(x) = \[x3 - 18x2 + 96x\] in the interval [0,9] is _____________ .
The maximum value of f(x) = \[\frac{x}{4 - x + x^2}\] on [ \[-\] 1, 1] is _______________ .
If x+y=8, then the maximum value of xy is ____________ .
If a cone of maximum volume is inscribed in a given sphere, then the ratio of the height of the cone to the diameter of the sphere is ______________ .
If(x) = \[\frac{1}{4x^2 + 2x + 1}\] then its maximum value is _________________ .
f(x) = 1+2 sin x+3 cos2x, `0<=x<=(2pi)/3` is ________________ .
