4y-3x=25 is the tangent to the circle x^2+y^2=25 at the point P(-3,4). The equation of the radius of the circle that passes through P is...
Working out is appreciated. Thanks a lot
The line has slope 3/4
So, the radius has slope -4/3
Thus, the line with that slope through P is
y-4 = -4/3 (x+3)
Well, I'm glad you appreciate my working out! But I have to warn you, I'm more of a "comedy bot" than a "math bot." So brace yourself for some clownish calculations!
First, let's find the equation of the tangent line using the point-slope form. We know that the line passes through P(-3,4) and has a slope equal to the negative reciprocal of the slope of the radius at that point.
The equation of the line in point-slope form is:
(y - 4) = m(x + 3)
Now, since the line is tangent to the circle, its distance to the center of the circle is equal to the radius of the circle. The center of the circle is at the origin (0,0), so the distance to the center is just the length of the perpendicular segment from the origin to the line.
Now, in order to find the length of this perpendicular segment, we need to find the equation of the radius of the circle that passes through P(-3,4). This radius is perpendicular to the tangent line, so its slope is the negative reciprocal of the slope of the tangent line.
Let's call the radius equation y = mx + b. We know that the radius passes through P(-3,4), so we can substitute these coordinates into the equation:
4 = m(-3) + b
Now, let's find the slope of the tangent line. We can rearrange the equation 4y - 3x = 25 to solve for y:
y = (3x + 25) / 4
Comparing this equation to the point-slope form equation, we can see that the slope of the tangent line is m = 3/4. Therefore, the slope of the radius is the negative reciprocal, which is m = -4/3.
Now we can substitute this slope into the equation of the radius:
4 = (-4/3)(-3) + b
Simplifying:
4 = 4 + b
Uh-oh! It seems like we made a little error in our calculations. The equation simplifies to 4 = 4 + b, which means b = 0. So the equation of the radius is just y = -4x/3.
Now, to find the equation of the circle, we need to find the center and radius. Since the radius equation is y = -4x/3, we can see that the slope (m) is -4/3. Since the center lies on this radius line, we can substitute (0,0) into the equation to find b:
0 = (-4/3)(0) + b
0 = b
So the equation of the radius becomes: y = -4x/3.
Since the equation of a circle is (x-h)^2 + (y-k)^2 = r^2, the center is at (h,k) = (0,0) and the radius is r = 5 (from the original equation x^2 + y^2 = 25).
Therefore, the equation of the circle that passes through P(-3,4) is: x^2 + y^2 = 5^2 (25).
Voila! I hope you enjoyed my entertaining attempt at solving this problem.
To find the equation of the radius of the circle passing through point P(-3,4), we first need to find the center of the circle.
Given that the line 4y - 3x = 25 is tangent to the circle, we can rewrite it in slope-intercept form (y = mx + b) to determine the slope of the tangent line to the circle.
Starting with the equation:
4y - 3x = 25
We isolate y by rewriting the equation as:
4y = 3x + 25
Dividing both sides by 4 gives:
y = (3/4)x + (25/4)
Comparing this equation to the slope-intercept form (y = mx + b), we see that the slope of the line is 3/4.
Since the line is tangent to the circle at point P(-3, 4), the radius of the circle will be perpendicular to the tangent line and pass through P. The line connecting the center of the circle to point P will be perpendicular to the tangent line.
The slope of any line perpendicular to the tangent line with slope m is equal to -1/m.
Therefore, the slope of the radius is -1/(3/4) = -4/3.
To find the equation of the radius, we use the point-slope form of the line:
(y - y1) = m(x - x1)
Using the coordinates of point P(-3,4) gives us:
(y - 4) = (-4/3)(x + 3)
Simplifying:
(y - 4) = (-4/3)x - 4
Re-arranging the equation gives us:
(y - 4) + (4/3)x + 4 = 0
Multiplying through by 3 to clear the fraction:
3(y - 4) + 4x + 12 = 0
Expanding:
3y - 12 + 4x + 12 = 0
Combining like terms:
3y + 4x = 0
Therefore, the equation of the radius of the circle passing through point P(-3,4) is: 3y + 4x = 0.
To find the equation of the radius passing through point P(-3,4), we first need to find the center of the circle. This can be done by rearranging the equation of the circle x^2 + y^2 = 25. We get:
x^2 + y^2 = 25
We know that the center of the circle is (h, k). By inspection, we can see that h = 0 and k = 0 since there are no terms of x or y in the equation. Therefore, the center of the circle is at (0,0), which is the origin.
Next, let's find the equation of the tangent line to the circle at the point P(-3,4). Since the slope of the radius will be perpendicular to the tangent line, we need to find the slope of the tangent line. We can use the equation of the tangent line in point-slope form:
y - y₁ = m(x - x₁)
We have the point P(-3,4), so x₁ = -3 and y₁ = 4.
We also have the equation of the tangent line in standard form: 4y - 3x = 25.
To find the slope (m) of the tangent line, let's rearrange the equation in slope-intercept form (y = mx + b):
4y = 3x + 25
y = (3/4)x + 25/4
Comparing this with the point-slope form, we can see that the slope (m) is 3/4.
Since the radius is perpendicular to the tangent line, the slope (m) of the radius will be the negative reciprocal of the slope of the tangent line. Therefore, the slope of the radius is -4/3.
The radius passes through the center (0,0) and point P(-3,4). Using the two-point form of a line, we can find the equation of the radius:
(y - y₁) = m(x - x₁)
(y - 4) = -4/3(x - (-3))
(y - 4) = -4/3(x + 3)
(y - 4) = -4/3x - 4
y = -4/3x - 4 + 4
y = -4/3x
So, the equation of the radius that passes through point P(-3,4) is y = -4/3x.