5. Consider the circle x2 + y2 = 16 and the parabola y2 = 8x. They intersect at P and Q in the first the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S.
a). The ratio of the areas of the triangles PQS and PQR is?
b). The radius of the circumcircle of the triangle PRS is?
c). The radius of the in-circle of the triangle PQR is?
To solve this problem, we will follow these steps:
Step 1: Finding the coordinates of points P and Q
- Set the equations of the circle and parabola equal to each other:
x^2 + y^2 = 16 --> (1)
y^2 = 8x --> (2)
- Substitute (2) into (1):
x^2 + (8x) = 16
- Simplify the equation:
x^2 + 8x - 16 = 0
- Solve the equation to find the x-coordinates of P and Q by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
where a = 1, b = 8, and c = -16
- Calculate the discriminant:
discriminant = b^2 - 4ac = (8)^2 - 4(1)(-16) = 64 + 64 = 128
- Substitute the values into the quadratic formula:
x = (-8 ± √(128)) / 2
= (-8 ± √(64 * 2)) / 2
= (-8 ± 8√2) / 2
= -4 ± 4√2
- So, we have two possible values for x: -4 + 4√2 and -4 - 4√2.
- Substitute these x-values back into equation (2) to find the corresponding y-coordinates:
For x = -4 + 4√2, y^2 = 8(-4 + 4√2)
= -32 + 32√2
For x = -4 - 4√2, y^2 = 8(-4 - 4√2)
= -32 - 32√2
- Taking the square root on both sides, choose the positive square roots since we are interested in points in the first and fourth quadrants:
For x = -4 + 4√2, y = √(-32 + 32√2)
For x = -4 - 4√2, y = √(-32 - 32√2)
Step 2: Finding the coordinates of points R and S
- Tangents to the circle at points P and Q intersect the x-axis at points R and S. The x-coordinate of the tangent point is the x-coordinate of P and Q, and the y-coordinate is 0 since it lies on the x-axis. Therefore, the coordinates of R and S are (x, 0).
Step 3: Finding the ratio of the areas of triangles PQS and PQR
- We can use the distance formula to find the lengths of the sides of triangles PQS and PQR.
- The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
- Calculate the distances:
PR = √((-4 + 4√2 - x)^2 + y^2)
PS = √((-4 + 4√2 - x)^2 + 0^2)
- The ratio of areas is given by:
Ratio = (Area PQS) / (Area PQR) = (PR * PS) / (PR * QR)
Step 4: Finding the radius of the circumcircle of triangle PRS
- The circumcircle of a triangle is the circle that passes through all three vertices of the triangle.
- The radius of the circumcircle can be found using the formula:
R = (abc) / (4A)
where a, b, c are the sides of the triangle, and A is the area of the triangle.
Step 5: Finding the radius of the in-circle of triangle PQR
- The in-circle of a triangle is the circle that is tangent to all three sides of the triangle.
- The radius of the in-circle can be found using the formula:
r = A / s
where A is the area of the triangle, and s is the semiperimeter (half the perimeter) of the triangle.
I will now calculate the values for you.
To answer these questions, we need to find the coordinates of points P, Q, R, and S. Let's start step by step:
Step 1: Find the coordinates of points P and Q
Substitute the equations of the circle and the parabola to find the intersection points.
From the equation of the circle, we have: x^2 + y^2 = 16.
From the equation of the parabola, we have: y^2 = 8x.
Substituting y^2 from the parabola equation into the circle equation, we get:
8x = 16 - x^2.
Rearranging the equation, we have:
x^2 + 8x - 16 = 0.
Solving this quadratic equation, we find two possible x-values for points P and Q.
Step 2: Find the y-values of points P and Q
Substitute the x-values obtained in step 1 into the equation of the parabola to find the corresponding y-values.
Step 3: Find the coordinates of point R
We know that point R lies on the tangent to the circle at point P and intersects the x-axis.
To find its x-coordinate, we need to find the slope of the tangent at point P by taking the derivative of the circle equation with respect to x:
dy/dx = -x/y.
Substituting the coordinates of point P into the derivative, we find the slope. With the slope and the given point, we can find the equation of the tangent line. Setting y equal to zero, we can solve for the x-coordinate of point R.
Step 4: Find the coordinates of point S
We can use the same process as step 3, but this time using the parabola equation and the point Q.
Once we have the coordinates of points P, Q, R, and S, we can answer the following questions:
a) To find the ratio of the areas of triangles PQS and PQR, you can use the formula for the area of the triangle: Area = (1/2) * base * height. In this case, PQ is the base and the heights can be found by subtracting the y-coordinates of P and Q from the x-axis. By calculating the areas of both triangles, you can find their ratio.
b) The radius of the circumcircle of the triangle PRS can be found using the Distance Formula. Calculate the distance between any two points among P, R, and S. Then, divide the distance by 2 to get the radius of the circumcircle.
c) To find the radius of the in-circle of triangle PQR, you can use the formula: r = Area / (1/2 * (Perimeter)). Calculate the area of the triangle using the formula from part a), and then calculate the perimeter by finding the lengths of the sides PQ, QR, and RP.