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Question
Answer the following :
Find the value of k, if the length of the tangent segment from the point (8, –3) to the circle
x2 + y2 – 2x + ky – 23 = 0 is `sqrt(10)`
Solution
The equation of the circle is
S ≡ x2 + y2 – 2x + ky – 23 = 0
Length of the tangent from (8, −3)
= `sqrt("S"_1)=sqrt(x_1^2 + y_1^2 - 2x_1 + "ky"_1 - 23)`, .........(where x1 = 8, y1 = –3)
= `sqrt(8^2 + (-3)^2 -2(8) + "k"(-3) -23)`
= `sqrt(64 + 9 - 16 - 3"k" - 23)`
= `sqrt(34 - "k")`
This is given to be `sqrt(10)` units
∴ `sqrt(34 - 3"k") = sqrt(10)`
∴ 34 – 3k = 10
∴ 3k = 24
∴ k = 8
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