Manipal Entrance Test 2013 Math Paper

The given circle
S(x,y)≡x2+y2−x−y−6=0 ...(i)
has centre at C ≡ (

1

2

,

1

2

) .
According to the given conditions, the given pointP (α - 1, α + 1) must lie inside the given circle.
i.e., S(α−1,α+1)<0
⇒ (α−1)2+(α+1)2−(α−1) −(α+1)−6<0
⇒ α2−α−2<0 i.e., (α−2)(α+1)<0
⇒ −1<α<2
[using sign - scheme from algebra] ........ (ii)
and also P and C must lie on the same side of the line. L(x,y)=x+y−2=0
i.e., L(

1

2

,

1

2

) and L(α−1,α+1) must have the same sign. Now, Isince L(

1

2

,

1

2

)=

1

2

+

1

2

−2<0
therefore, we have
L(α−1,α+1)=(α−1) +(α+1)−2<0
⇒ α<1 ......(iv)
Ineqalities (ii) and (iv) together give the permissible values of α as - 1 < α < 1.