Question

The minimum distance between the circle \( x^{2}+y^{2}=9 \) and the curve \( 2 x^{2}+10 y^{2}+6 x y=1 \) is 

1) \( 2 \sqrt{2} \) 

2) 2 

3) \( 3-\sqrt{2} \) 

4) \( 3-\frac{1}{\sqrt{11}} \)

Answer 1

Correct option is 2) 2

x² + y² = 9....(i)

centre of circle is (0,0)

2x² + 10y² + 6xy = 1....(ii)

Partial differentiate with respect to x

4x + 6y = 0

Partial differentiate with respect to y

20y + 6x = 0

Solution of above given (x,y) = (0,0)

Hence centre of (ii) is also (0,0)

Also shortest distance is always found along common normal, and normal of circle passes through centre (0,0), hence we need shortest distance of origin from any point (x,y) on (ii) is

Now you to minimize x² + y² subjected to condition 

2x² + 10y² + 6xy = 1 which will given d = 1

Hence

Shortest distance = Radius −d = 3 − 1 = 2