Mathematics Times – July 2019

SECTION-
5.Sol:

meets the parabola y x^2 at twopoints

B 1,1 and also at F  1,1.

The region, say R 1 bounded by the three curves

is a part of the given set. It is easier obtain its

area, say by horizontal than by vertical

slicing. On R 1 , y varies from 1 to 4. For each

y 0  1, 4 , the intersection of R 1 with the

liney y 0 is the segment from  y 0 ,1to

0

8

, 0

y

 

 

 

. It is shown by DE in the figure

above. Its length is^0

0

8

y

y

. Hence

4

0

(^10)
8
y dy
y
 
    
 



3 4

2

1

2

(8 log )

3

y

e

y

y y





 

16 2

8log 4

e 3 3

  

14

16 log 2

e 3

 

So A is correct.

SECTION-
5.Sol: Given an ellipse
2 2
: 2 2 1

x y

E

a b

  (1)

let us first identify the rectangle, say R,of

maximum area inscribed in E and sides parallel

to the axes of E. If a bcos , sin  is its

vertex in the first quadrant, the other vertices

are obtained by taking reflections into the axes.

So the sides of this rectangle are 2 cosa 

and 2 sinb . Hence its area is

4 absin cos  2 ab 2  which is maximum

when

4



.

So, the rectangle R of maximum area has its

vertices at ,

2 2

 a b 

  

 

.

Next, we have to find an ellipse E'with sides

parallel to those of R and having the largest

area. Clearly, the marjor and minor axes

ofE'must be the lengthand the width of R,

i.e. 2 a and 2 brespectively. So,

2 2

' : 2 2 1

2 2

x y

E

a b

 

(2)

This construction of interlacing ellipses and

rectangles then goes on. The figure above

shows the construction ofR 1 from E 1 and

E 2 fromR 1.

The crucial observation is that the lengths of

the semi-major and semi-minor axes each form

a geometric progression with common

ratio

1

2

. We have a b 1  3, 1 2. Hence for

the n th ellipse En,we have

 

1

3

2

an n (3)

and  ^1

2

2

bn n (4)

Note that the ratio of the major to the minor

axis of each ellipse is the same, as follows

3

2

.

Since the eccentricity depends only on this

ratio, (A) is immediately ruled out. Further this