Question

Find the length of tangent drawn from any point on circle x square + y square + 4 x + 6 y - 3 to the circle x square + y square + 4 x + 6 y + 4 = 20

Answer 1

Let PQ is a tangent drawn from point P on circle x² + y² + 4x + 6y - 3 = 0 to the circle x² + y² + 4x + 6y + 4 = 20 or x² + y² + 4x + 6y - 16 = 0.

here, you should notice that given circles are concentric circles because centre of both circles are same e.g., (-2, -3).

arrangement is shown in figure,

radius of inner circle, r = √(g² + f² - c) = √{(-2)² + (-3)² - (-3)} = √(4 + 9 + 3) = 4

radius of outer radius , R = √{(-2)² +(-3)² - (-16)} = √(4 + 9 + 16)}= √(29)

from Pythagoras theorem,

PQ² = R² - r² = (√29)² - 4² = 29 - 16 = 13

hence, PQ = √13 unit

so, length of tangent is √13 unit.