Verify that the circles x2+y2 = 25 and (x−5)2+(y−10)2 = 50 intersect at
A = (4, 3).
Find the size of the acute angle formed at A by the intersecting circles. You will first have
to decide what is meant by the phrase the angle formed by the intersecting circles.
To verify that the circles intersect at point A = (4, 3), we need to substitute these coordinates into the equations of the circles and see if they satisfy both equations.
For the first circle, x^2 + y^2 = 25:
(4)^2 + (3)^2 = 16 + 9 = 25
So, point A lies on the first circle.
For the second circle, (x-5)^2 + (y-10)^2 = 50:
(4-5)^2 + (3-10)^2 = 1 + 49 = 50
So, point A lies on the second circle as well.
Therefore, we have verified that both circles intersect at point A = (4, 3).
Now, to find the size of the acute angle formed at A by the intersecting circles, we need to understand what is meant by "the angle formed by the intersecting circles."
The angle formed by the intersecting circles refers to the angle formed by the tangent lines at the point of intersection. In other words, we are looking for the angle between the tangent lines to the circles at point A.
To find the tangent lines, we need to find the derivatives of the equations of the circles.
For the first circle, x^2 + y^2 = 25:
Differentiating both sides with respect to x:
2x + 2y(dy/dx) = 0
dy/dx = -x/y
For the second circle, (x-5)^2 + (y-10)^2 = 50:
Differentiating both sides with respect to x:
2(x-5) + 2(y-10)(dy/dx) = 0
dy/dx = (5-x)/(y-10)
Now, let's find the slopes of the tangent lines at point A.
Substituting the coordinates of point A into the derivatives:
For the first circle:
dy/dx = -4/3
For the second circle:
dy/dx = (5-4)/(3-10) = 1/-7 = -1/7
Now, we can find the angle formed by the intersecting circles using the formula:
tan(angle) = |(m2 - m1) / (1 + m1*m2)|
Where m1 and m2 are the slopes of the tangent lines.
Plugging in the values:
tan(angle) = |(-1/7 - (-4/3)) / (1 + (-4/3)*(-1/7))|
= |-1/7 + 4/3| / (1 + 4/21)
= |(-21 + 28) / (21 + 4)| / (25/21)
= |7/25| / (25/21)
= (7/25) / (25/21)
= 7/25 * 21/25
= 147/625
Now, to find the actual angle from the tangent, we take the inverse tangent (arctan) of the value obtained:
angle = arctan(147/625)
Using a calculator, we find that the angle is approximately 13.91 degrees.
Therefore, the size of the acute angle formed at A by the intersecting circles is approximately 13.91 degrees.
To verify that the circles intersect at point A = (4, 3), we can substitute the coordinates of A into both circle equations and check if they are satisfied.
For the first circle, substituting x = 4 and y = 3, we get:
(4)^2 + (3)^2 = 25
16 + 9 = 25
25 = 25
So, point A = (4, 3) lies on the first circle.
Now, for the second circle, substituting x = 4 and y = 3, we get:
(4-5)^2 + (3-10)^2 = 50
(-1)^2 + (-7)^2 = 50
1 + 49 = 50
50 = 50
So, point A = (4, 3) lies on the second circle as well.
Since point A satisfies both circle equations, we can conclude that the circles intersect at point A = (4, 3).
Next, let's understand what is meant by the "angle formed by the intersecting circles." In this context, the intersecting circles create two points of intersection, which leads to a chord connecting these two points. This chord creates an angle with the center of each circle, which we will refer to as the "central angle."
Now, to find the size of the acute angle formed at A by the intersecting circles, we need to calculate the measure of the central angle at point A.
To find the central angle at point A, we can use the properties of circles. The central angle is equal to twice the inscribed angle that subtends the same arc.
The chord connecting the points of intersection divides the circle into two arcs. The arc opposite to point A will be subtended by the acute angle formed by the intersecting circles at point A.
To calculate the measure of this inscribed angle, we need to find the lengths of the two chords connecting the intersecting points to point A and then apply the inscribed angle theorem.
Let's find the lengths of the chords first. The distance between points A = (4, 3) and the center of the first circle (0, 0) is given by:
√((x2 - x1)^2 + (y2 - y1)^2)
= √((4 - 0)^2 + (3 - 0)^2)
= √(4^2 + 3^2)
= √(16 + 9)
= √25
= 5
Similarly, the distance between points A = (4, 3) and the center of the second circle (5, 10) is given by:
√((x2 - x1)^2 + (y2 - y1)^2)
= √((4 - 5)^2 + (3 - 10)^2)
= √((-1)^2 + (-7)^2)
= √(1 + 49)
= √50
Now, we have the lengths of the two chords, which are 5 and √50.
According to the inscribed angle theorem, the measure of the inscribed angle is equal to half the measure of its intercepted arc.
By using the properties of circles, we can find the relation between the length of a chord (c) and the measure of the inscribed angle (θ) as:
θ = 2 * arcsin(c/2r)
For the first circle:
θ1 = 2 * arcsin(5/2 * √(25))
θ1 = 2 * arcsin(5/10)
θ1 = 2 * arcsin(1/2)
θ1 ≈ 2 * 30°
θ1 ≈ 60°
For the second circle:
θ2 = 2 * arcsin(√50/2 * √(50))
θ2 = 2 * arcsin(√50/10)
θ2 = 2 * arcsin(√5/2)
θ2 ≈ 2 * 45°
θ2 ≈ 90°
Note: The angle formed at A will be the smaller acute angle between the two circles. In this case, it is θ1 ≈ 60°.
Therefore, the size of the acute angle formed at A by the intersecting circles is approximately 60 degrees.