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.