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)`

Answer 1

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