KVPY SX SB Exam 2012 Question Paper

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Question : 83 of 120

Marks: +1, -0

Let (x, y) be a variable point on the curve

4x^2+9y^2-8x-36y+15=0

.Then min

(x^2 - 2x + y^ 2 - 4y + 5)

+ max

(x^2 - 2x + y^2 - 4y + 5)

is

$325/36$
$36/325$
$13/25$
$25/13$

Solution:

4x^2 + 9y^2 - 8x -36y + 15 = 0

4(x^2-2x)+9(y^2-4y)=-15

4(x^2-2x+1)

+9(y^2-4y+4)

=-15+4+36

4(x-1)^2+9(y-2)^2=25

(x-1)^2/(5/2)^2+ (y-2)^2/(5/3)^2=1

.............(1)

x^2-2x+y^2-4y+5

(x-1)^2+(y-2)^2

min of

((x-1)^2+(y-2)^2)=25/9

max of

((x-1)^2+(y-2)^2)=25/4

=25/9+25/4=325/36

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