NCERT Class 12-Mathematics: Chapter –6 Application of Derivatives Part 5

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Question 14:

Find the area of greatest rectangle that can be inscribed in an ellipse

Answer:

Let be the rectangle of maximum area with sides and , where is a point on the ellipse as shown in the Fig.6.3. The area A of the rectangle is i.e. which gives (say)

Therefore,

Again,

Now,

At

Thus at is maximum and hence the area A is maximum.

Maximum area units.

Question 17:

Find the difference between the greatest and least values of the function , on

Answer:

Therefore,

Clearly, is the greatest value and is the least.

Therefore, difference

Question 18:

An isosceles triangle of vertical angle is inscribed in a circle of radius . Show that the area of triangle is maximum when

Answer:

Let ABC be an isosceles triangle inscribed in the circle with radius a such that .

(see fig. 16.4)

Therefore, area of the triangle ABC i.e.

Therefore,

Therefore,

Therefore, Area of triangle is maximum when