First slide

Definition of a circle

Question

Point M moved on the circle $(x - 4)^{2} + (y - 8)^{2} = 20$ .Then it broke away from it and moving along a tangent to the circle cut the x-axis at point (-2,0). The coordinates of the point on the circle at which the moving point broke away is

Moderate

A

$(42 / 5, 36 / 5)$
B

$(- 2 / 5, 44 / 5)$
C

$(6, 4)$
D

$(2, 4)$

Solution

$x^{2} + y^{2} - 8 x - 16 y + 60 = 0 - - - - - (i) Equation of chord of contact from (- 2, 0) is - 2 x - 4 (x - 2) - 8 y + 60 = 0 or 3 x + 4 y - 34 = 0 - - - - - - (ii) Solving (i) and (ii), we get x^{2} + {(\frac{34 - 3 x}{4})}^{2} - 8 x - 16 (\frac{34 - 3 x}{4}) + 60 = 0 or 16 x^{2} + 1156 - 204 x + 9 x^{2} - 128 x - 2176 + 192 x + 960 = 0 or 5 x^{2} - 28 x - 12 = 0 or (x - 6) (5 x + 2) = 0 or x = 6, - \frac{2}{5} Therefore, the points are (6, 4) and (- 2 / 5, 44 / 5) .$

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