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

NA

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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