Question 264572: the circle, centre (p,q) radius 25, meets the x axis at (-7,0) and (7,0) where q>o.
find the value of p and q
find the coordinates of the points where the circle meets the y axis
THANK YOU SO MUCH IF YOU CAN HELP!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! formula for the circle is (x-p)^2 + (y-q)^2 = 25^2
this is from the standard formula of (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center of the circle, and r^2 is the square of the radius of the circle.
the center of your circle is (p,q), so we have h = p and q = k.
the radius of your circle is 25, which makes the equation:
(x-p)^2 + (y-q)^2 = 25^2
we know that (0,-7) is on the circle and (0,7) is on the circle.
we know that from the center of the circle to (0,-7), the length is 25.
we know that from the center of the circle to (0,7), the length is 25.
if we call the center of the circle A, and the point (0,-7) B, and the point (0,7) C, then we have an isosceles triangle whose sides are each 25 units in length and whose base is 14 units in length is sqrt (-7-7)^2 + (0-0)^2) = sqrt(14^2) = 14.
if we drop a perpendicular from the center of the circle (point A), perpendicular to the base of the triangle (side BC), then we have something that looks like 2 right triangles, with the x-axis as the base and the y-axis as the altitude.
we'll call the intersection of the perpendicular to the base point D.
The 2 right triangles are ABD and ACD.
side BD is 7 units in length.
side CD is 7 units in length.
side AB is 25 units in length.
side AC is 25 units in length.
We can find the length of AD by finding the angle ACD.
Cosine of ACD = adjacent / hypotenuse = CD / AC = 7/25 = .28
Arc Cosine of .28 = 73.73979529 degrees.
Tangent of 73.73979529 degrees = opposite / adjacent = AD / DC = AD / 7
multiply both sides of this equation by 7 to get:
7 * Tangent of 73.73979529 degrees = AD.
This makes AD = 24 units in length.
Since point A is the center of the circle and is on the y-axis 24 units above the x-axis, this makes the center of the circle equal to (x,y) = (0,24)
Since the center of the circle is (0,24), then the equation of the circle becomes:
(x-0)^2 + (y-24)^2 = 25^2
We can graph this equation by solving for y.
We solve for y to get y = 24 +/- sqrt(25^2 - x^2))
graph of that equation is shown below:
A diagram of what the circle looks like with it's isosceles triangle is shown below:
The y-axis is the vertical line in the middle.
The x-axis is the horizontal line at the 0 mark.
The question was:
find the value of p and q
find the coordinates of the points where the circle meets the y axis
The value of (p,q) = (0,24)
The points where the circle meets the y-axis are (0,49) and (0,-1)./
This is found by looking at the graph and solving the following equation.
The equation is x^2 + (y-24)^2 = 25^2
when x = 0, the equation becomes (y-24)^2 = 25^2
take square root of both sides of this equation to get y-24 = +/- 25
add 24 to both sides of this equation to get y = 24 +/- 25
24 + 25 = 49
24 - 25 = -1
|
|
|
