If the length of the tangent from (2, 5) to $$x^2 + y^2 - 5x + 4y + k = 0$$​ is $$\sqrt{37}$$​​ units, then the value of $$k$$​​ is:​

Correct option is C

Given:

circle is: x² + y² - 5x + 4y + k = 0

The external point is: (2, 5)

Length of the tangent: $$\sqrt{37}$$​​

Concept Used:

Tangent to a Circle: A tangent line to a circle intersects the circle at exactly one point.

Length of Tangent: The length of a tangent from an external point to a circle is the distance between that point and the point of tangency.

The length of the tangent from an external point (x₁, y₁) to the circle x² + y² + 2gx + 2fy + c = 0 is given by:$$\sqrt{(x₁² + y₁² + 2gx₁ + 2fy₁ + c)}$$​​

Solution:

In our case, the circle is: x² + y² - 5x + 4y + k = 0

The external point is: (2, 5)

Length of the tangent: $$\sqrt{37}$$​​

Substitute the values into the formula:$$\sqrt{(2² + 5² - 5(2) + 4(5) + k)}$$​ = $$\sqrt{37}$$​​

Square both sides of the equation:2² + 5² - 5(2) + 4(5) + k = 37

4 + 25 - 10 + 20 + k = 37

39 + k = 37

k = 37 - 39

k = -2

Therefore, the value of k is -2.

Option (c) is right.