Question

Passage: Let ∑ be a family of circles passing through the two points P(3, 7) and Q(6, 5).

Answer the following questions.

(i) The number of circles belonging to ∑ and touching the x-axis is

(A)  1 

(B)  2 

(C)  infinite

(D)  0

(ii) If each of the circles of ∑ cuts the circle x² + y² - 4x -  6y -   3 = 0, then all these chords pass through a fixed point whose coordinates are

(a) (2 + 23/3)

(b)  (-3 + 23/3)

(c)  (1,23)

(d)  (23,1)

(iii)  The centre of the circle belonging to ∑ and cutting orthogonally the circle x² + y² = 29 is

(a)  (1/2,3/2)

(b)  (7/2,9/2)

(c)  (3,-7/9)

(d)  (65/18,14/3)

Answer 1

Correct option(i)(b)(ii)(a)(iii)(d)

Explanation :

(i)  Equation of PQ is 2x + 3y -  27 = 0. Also the circle described on PQ is a diameter is S ≡ x² + y² - 9x - 12y + 53 = 0. Any circle passing through P and Q is of the form

 x² + y² - 9x - 12y + 53 + λ(2x + 3y - 27) = 0

 x² + y² - (9 - 2λ)x - (12 - 3λ) y + 53 -  27λ = 0 ....(1)

This touches the x-axis. That is

The discriminant of the quadratic is positive so that it has two distant roots. Hence, there are two circles belonging to ∑ which touch the x-axis.

 This chord passes through the intersection of  - 5x - 6y + 56 = 0 and 2x + 3y - 27 = 0 which is (2, 23/3).

(iii) The circle given in Eq. (1) cuts orthogonally the circle

 Therefore, the circle is

 Therefore, the centre is

(65/18, 14/3)