Question

22. For which value of a in equation x^2 + y^2 +
(a^2 - 4)xy + 2x + 2y + a = 0 represents circle?
(Gujarat CET-2007]
(b) 2
(d) None of these
(a) 3
(c) 4​

Answer 1

Answer:

Equation of the given circle can be written as

(x2+y2−2x−8)−2a(y)=0    

which represents a family of circles passing through the intersection of the circle x2+y2−2x−8=0 and the line y=0.

The circle and the line intersect at the points P(−2,0)  and Q(4,0).

Let the tangents at P and Q to a member of this family intersect at (h,k), 

then PQ is the chord of contact of (h,k) and its equation is

hx+ky−(x+h)−a(y+k)−8=0 ⇒x(h−1)+y(k−a)−(h+ak+8)=0

Comparing this with equation y=0 of PQ, we get 

h=1, h+ak+8=0, 

Since (h,k) lies on the given line

 x+2y+5=0 1+2k+5=0⇒k=−3 and 1−3a+8=0⇒a=3

Hence the equation of the required member C of the family is 

x2+y2−2x−6y−8=0 

Step-by-step explanation:

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