To find the equation of the line PQ, we'll use the following steps:
1. Find the center of the circle P.
2. Find the radius of the circle P.
3. Use the center and radius to find the equation of the circle.
4. Find the equation of the line PQ, which is the tangent line to the circle at point Q.
Let's go through these steps one by one:
1. Find the center of the circle P:
Since the center of the circle P is on the line x + y + 7 = 0, we can rewrite this equation in the form y = -x - 7. This equation represents a line with a slope of -1 and a y-intercept of -7. Therefore, the center of the circle P is at a point where x = 0 and y = -7. So, the center of the circle P is (0, -7).
2. Find the radius of the circle P:
We know that line 3x - 4y = 4 is a tangent line to the circle at point Q (-4, -4). The distance from the center of the circle P to point Q is equal to the radius of the circle.
Using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((-4 - 0)^2 + (-4 - (-7))^2)
Distance = √(16 + 9)
Distance = √25
Distance = 5
So, the radius of the circle P is 5.
3. Use the center and radius to find the equation of the circle:
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values:
(x - 0)^2 + (y - (-7))^2 = 5^2
x^2 + (y + 7)^2 = 25
4. Find the equation of the line PQ:
Now that we have the equation of the circle P, we can find the equation of the tangent line PQ. The tangent line at a point on a circle is perpendicular to the radius at that point. So, the slope of the line PQ is the negative reciprocal of the slope of the radius from the center (0, -7) to Q (-4, -4).
Slope of radius = (y2 - y1) / (x2 - x1) = (-4 - (-7)) / (-4 - 0) = 3 / (-4) = -3/4
The slope of the tangent line PQ is the negative reciprocal of -3/4, which is 4/3.
Using the point-slope form of a line with the point Q (-4, -4):
y - y1 = m(x - x1)
y - (-4) = (4/3)(x - (-4))
y + 4 = (4/3)(x + 4)
Now, we can simplify and put it in standard form:
3(y + 4) = 4(x + 4)
Expand and rearrange:
3y + 12 = 4x + 16
Subtract 12 from both sides:
3y = 4x + 16 - 12
3y = 4x + 4
Divide by 3:
y = (4/3)x + 4/3
So, the equation of the line PQ is y = (4/3)x + 4/3.