Advertisements
Advertisements
प्रश्न
Find the area of greatest rectangle that can be inscribed in an ellipse `x^2/"a"^2 + y^2/"b"^2` = 1
Advertisements
उत्तर
Let ABCD be the rectangle of maximum area with sides AB = 2x and BC = 2y
Where C (x, y) is a point on the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 as shown in the Fig.6.3.
The area A of the rectangle is 4xy
i.e. A = 4xy which gives A2 = 16x2y2 = s ....(say)
Therefore, s = `16x^2 (1 - x^2/"a"^2)*"b"^2`
= `(16"b"^2)/"a"^2 ("a"^2x^2 - x^4)`
⇒ `"ds"/"dx" = (16"b"^2)/"a"^2 * [2"a"^2x - 4x^3]`
Again, `"ds"/"dx"` = 0
⇒ x = `"a"/sqrt(2)` nad y = `"b"/sqrt(2)`
Now, `("d"^2"s")/("dx"^2) = (16"b"^2)/"a"^2 [2"a"^2 - 12x^2]`
At x = `"a"/sqrt(2), ("d"^2"s")/("dx"^2)`
= `(16"b"^2)/"a"^2 [2"a"^2 - 6"a"^2]`
= `(16"b"^2)/"a"62 (-4"a"^2) < 0`
Thus at x = `"a"/sqrt(2)`, y = `"b"/sqrt(2)`
s is maximum and hence the area A is maximum.
Maximum area = 4.x.y
= `4 * "a"/sqrt(2) * "b"/sqrt(2)`
= 2ab sq units.
APPEARS IN
संबंधित प्रश्न
A cone is inscribed in a sphere of radius 12 cm. If the volume of the cone is maximum, find its height
f (x) = x2/3 on [−1, 1] Discuss the applicability of Rolle's theorem for the following function on the indicated intervals ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 1)2 on [0, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = x(x − 4)2 on the interval [0, 4] ?
Verify Rolle's theorem for each of the following function on the indicated interval f (x) = cos 2 (x − π/4) on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = sin 2x on [0, π/2] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = ex sin x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = cos 2x on [0, π] ?
Verify Rolle's theorem for the following function on the indicated interval f (x) = log (x2 + 2) − log 3 on [−1, 1] ?
Verify Rolle's theorem for the following function on the indicated interval f(x) = 2 sin x + sin 2x on [0, π] ?
Using Rolle's theorem, find points on the curve y = 16 − x2, x ∈ [−1, 1], where tangent is parallel to x-axis.
At what point on the following curve, is the tangent parallel to x-axis y = \[e^{1 - x^2}\] on [−1, 1] ?
At what point on the following curve, is the tangent parallel to x-axis y = 12 (x + 1) (x − 2) on [−1, 2] ?
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem f(x) = x2 − 2x + 4 on [1, 5] ?
Discuss the applicability of Lagrange's mean value theorem for the function
f(x) = | x | on [−1, 1] ?
Show that the lagrange's mean value theorem is not applicable to the function
f(x) = \[\frac{1}{x}\] on [−1, 1] ?
Find a point on the curve y = x2 + x, where the tangent is parallel to the chord joining (0, 0) and (1, 2) ?
Find a point on the parabola y = (x − 3)2, where the tangent is parallel to the chord joining (3, 0) and (4, 1) ?
Find the points on the curve y = x3 − 3x, where the tangent to the curve is parallel to the chord joining (1, −2) and (2, 2) ?
If f (x) = Ax2 + Bx + C is such that f (a) = f (b), then write the value of c in Rolle's theorem ?
State Rolle's theorem ?
If the value of c prescribed in Rolle's theorem for the function f (x) = 2x (x − 3)n on the interval \[[0, 2\sqrt{3}] \text { is } \frac{3}{4},\] write the value of n (a positive integer) ?
Find the value of c prescribed by Lagrange's mean value theorem for the function \[f\left( x \right) = \sqrt{x^2 - 4}\] defined on [2, 3] ?
If f (x) = ex sin x in [0, π], then c in Rolle's theorem is
Show that the local maximum value of `x + 1/x` is less than local minimum value.
Find the difference between the greatest and least values of the function f(x) = sin2x – x, on `[- pi/2, pi/2]`
Prove that f(x) = sinx + `sqrt(3)` cosx has maximum value at x = `pi/6`
What does continuity on a closed interval guarantee?
For a continuous function on the closed interval \[a,d\], what does \[f(b)\] represent?
Which statement about local extrema and absolute extrema is correct?
Which points must always be checked when finding absolute extrema on a closed interval \[a,b\]?
How are the absolute maximum and absolute minimum determined after evaluating \[f(x)\] at all critical points and endpoints?
For \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\], which critical point is obtained from \[2(8x-1)=0\]?
What is \[f(-1)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What is \[f\left(\frac{1}{8}\right)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What is \[f(0)\] for \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\]?
What are the absolute extrema of \[f(x)=12x^{\frac{4}{3}}-6x^{\frac{1}{3}}\] on \[-1,1\]?
