if a diameter of the circle 2x^2 +2y^2+4x+ky-27=0 passes through the point5,5 and the slope of the diameter is 0.5 find the value of k

Step 1: Rewrite the equation of the circle in standard form

The given equation of the circle is 2x^2 + 2y^2 + 4x + ky - 27 = 0. We can rewrite this in the standard form (x-h)^2 + (y-k)^2 = r^2 by completing the square.

First, divide the entire equation by 2 to simplify:

x^2 + y^2 + 2x + (k/2)y - 13.5 = 0

Then, rearrange the terms:

(x^2 + 2x) + (y^2 + (k/2)y) = 13.5

To complete the square, we add and subtract (b/2)^2 inside each bracket, where b is the coefficient of x in the first bracket and the coefficient of y in the second bracket:

[(x^2 + 2x + 1) - 1] + [(y^2 + (k/2)y + (k/4)^2) - (k/4)^2] = 13.5

This simplifies to:

(x + 1)^2 + (y + k/4)^2 = 13.5 + 1 + (k/4)^2

So, the standard form of the circle's equation is:

(x + 1)^2 + (y + k/4)^2 = 14.5 + (k/4)^2

Step 2: Use the given point and slope to find k

The diameter of the circle passes through the point (5,5) and has a slope of 0.5. The diameter of a circle is a line segment that passes through the center of the circle. Therefore, the center of the circle is the midpoint of the diameter. Since the slope of the diameter is 0.5, the line perpendicular to the diameter (which passes through the center of the circle) has a slope of -2 (because the product of the slopes of two perpendicular lines is -1).

The equation of the line passing through the point (5,5) with a slope of -2 is:

y - 5 = -2(x - 5)

Solving for y gives:

y = -2x + 15

This line passes through the center of the circle, so we can substitute y = -2x + 15 into the equation of the circle:

(x + 1)^2 + (-2x + 15 + k/4)^2 = 14.5 + (k/4)^2

Solving this equation for k will give the value of k.