Find the equation of the family of circles which are passing through the points (1, 3) and (4, 5) ?

CONCEPT:

The equation of family of circles passing through two points say (x1, y1) and (x2, y2) is given by:\(\left( {x - {x_1}} \right) ⋅ \left( {x - {x_2}} \right) + \left( {y - {y_1}} \right) ⋅ \left( {y - {y_2}} \right) + λ ⋅ \left| {\begin{array}{*{20}{c}} x&y&1\\ {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1 \end{array}} \right| = 0\) where λ ≠ - 1.

CALCULATION:

Here, we have to find the equation of the family of circles which passing through the points (1, 3) and (4, 5)

As we know that, the equation of family of circles passing through two points say (x1, y1) and (x2, y2) is given by:\(\left( {x - {x_1}} \right) ⋅ \left( {x - {x_2}} \right) + \left( {y - {y_1}} \right) ⋅ \left( {y - {y_2}} \right) + λ ⋅ \left| {\begin{array}{*{20}{c}} x&y&1\\ {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1 \end{array}} \right| = 0\)

Here, x1 = 1, y1 = 3, x2 = 4 and y2 = 5.

So, the required equation is: \(\left( {x - {1}} \right) ⋅ \left( {x - {4}} \right) + \left( {y - {3}} \right) ⋅ \left( {y - {5}} \right) + λ ⋅ \left| {\begin{array}{*{20}{c}} x&y&1\\ {{1}}&{{3}}&1\\ {{4}}&{{5}}&1 \end{array}} \right| = 0\)

⇒ (x2 - 5x + 4) + (y2 - 8y + 15) + λ ⋅ (- 2x + 3y - 7) = 0

⇒ x2 + y2 - (5 + 2λ) x + (3λ - 8)y + (19 - 7λ) = 0

Hence, option A is the correct answer.