from a point P(2,3) tangents PA,PB are drawn to the circle x^2+y^-6x+8y-1=0.The equation of the line joining the mid points of PA and PB is?
Answers
Given : from a point P(2,3) tangents PA,PB are drawn to the circle x²+y²-6x+8y-1=0.
To Find : The equation of the line joining the mid points of PA and PB is?
Solution:
x²+y²-6x+8y-1=0
=> (x - 3)² - 9 + (y + 4)² - 16 - 1 =0
=> (x - 3)² + (y + 4)² = 26
=> (x - 3)² + (y + 4)² = (√26)²
center O( 3 , - 4)
radius = √26
P = ( 2, 3)
PA² + OA² = OP² , PB² + OB² = OP²
OP² = (3 - 2)² + (-3 - 4)² = 50
OA² = OB² = 26
PA² = PB² = 24
(x - 2)² + (y - 3)² = 24
(x - 3)² + (y + 4)² = 26
=> -2x + 5 + 14y + 7 = 2
=> -2x + 14y = -10
=> -x + 7y = - 5
=> x = 7y + 5
(7y + 5 - 3)² + (y + 4)² = 26
=> (7y + 2)² + (y + 4)² = 26
=> 49y² + 28y + 4 + y² + 8y + 16 = 26
=> 50y² + 36y - 6 =0
=> 25y² + 18y - 3 = 0
=> y =( - 18 ± 24.98)/50
=> y = 0.1396 , -0.8596
x = 5.9772 , -1.0172
( 5.9772 , 0.1396 ) and ( -1.0172 , -0.8596)
P = ( 2 , 3)
mid point of PA & PB
( 3.9886 , 1.5698) and (0.4914 , 1.0702)
(3.99 , 1.57) and (0.49 , 1.07)
Slope = ( -0.5/-3.5) = 1/7
y - 1.57 = (1/7)(x - 3.99)
=> 7y - 10.99 = x - 3.99
=> 7y = x + 7
equation of the line joining the mid points of PA and PB
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