Answer check please: Consider the circle (x-3)^2 + (y+1)^2 = 169. The point (15,4) is on the circle. Find an equation for the line that is tangent to the circle at this point. I got y= 5/12x-9/4.
C(3,-1), P(15,4).
m1 = (4+1) / (15-3) = 5/12.
m2 = -12/5 = Negative reciprocal 0f m1.
P(15,4).
Y = mx + b = 4.
(-12/5)15 + b = 4
-36 + b = 4
b = 40.
Eq: Y = (-12/5)x + 40.
what angle does an arc 6.6cm in length substend at the centre of a circle of radius 14cm if ^=22/7
To check if your answer is correct, we can start by finding the slope of the tangent line.
Given the equation of the circle: (x - 3)^2 + (y + 1)^2 = 169, we can rewrite it as:
(x - 3)^2 = 169 - (y + 1)^2
We know that the point (15, 4) lies on the circle, so we can substitute these values into the equation:
(15 - 3)^2 = 169 - (4 + 1)^2
12^2 = 169 - 5^2
144 = 169 - 25
144 = 144
Since the equation is true, the point (15, 4) does lie on the circle.
The slope of the tangent line can be determined by taking the derivative of the equation of the circle with respect to x. Differentiating both sides of the equation, we get:
2(x - 3) = -2(y + 1) * (dy/dx)
At the point (15, 4), we can substitute the x and y coordinates into this equation:
2(15 - 3) = -2(4 + 1) * (dy/dx)
24 = -10 * (dy/dx)
(dy/dx) = -24/10
(dy/dx) = -12/5
The slope of the tangent line is -12/5.
Now, to find the equation of the tangent line, we know that the equation of a line can be written as:
y - y₁ = m(x - x₁)
Where (x₁, y₁) is a point on the line and m is the slope. We can use the point (15, 4) and the slope -12/5 to substitute into this equation:
y - 4 = (-12/5)(x - 15)
Simplifying the equation gives:
y - 4 = (-12/5)x + 36
Finally, rearranging the equation to the standard form, we get:
(12/5)x + y = 40/5
12x + 5y = 40
So, the equation of the tangent line is 12x + 5y = 40.
Comparing this to your answer, y = (5/12)x - 9/4, we see that the slopes are reciprocal to each other, and therefore the equations are not the same.
Hence, your answer y = 5/12x - 9/4 is incorrect, and the correct equation for the line that is tangent to the circle at the point (15, 4) is 12x + 5y = 40.