Let c and dd be positive real numbers. Show that regardless of c,d, the parabolas r=c/(1+cos(theta)) and r=d/{(1-costheta) intersect at rights angles
To show that the parabolas intersect at right angles, we need to demonstrate that the tangent lines to the curves at the intersection point are perpendicular.
Let's find the equations of the tangent lines and verify their perpendicularity.
Step 1: Find the equations of the parabolas.
The equation of a parabola in polar coordinates is given by r = f(θ) where f(θ) represents a function. In this case, we have two parabolas:
Parabola 1: r₁ = c / (1 + cos(θ))
Parabola 2: r₂ = d / (1 - cos(θ))
Step 2: Find the derivative of each equation.
To find the slope of the tangent line at any point on the curve, we need to differentiate the equation of the curve with respect to θ.
Differentiating parabola 1:
dr₁/dθ = -c*sin(θ) / (1 + cos(θ))^2
Differentiating parabola 2:
dr₂/dθ = d*sin(θ) / (1 - cos(θ))^2
Step 3: Find the values of θ where the parabolas intersect.
To find the intersection points, we equate the equations of the parabolas:
c / (1 + cos(θ)) = d / (1 - cos(θ))
Now, solve for cos(θ):
c(1 - cos(θ)) = d(1 + cos(θ))
c - c*cos(θ) = d + d*cos(θ)
c - d = (c+d)*cos(θ)
cos(θ) = (c - d) / (c + d)
Step 4: Calculate the slopes of the tangent lines at the intersection point.
Substitute the value of cos(θ) into the derivatives of parabolas 1 and 2 obtained in Step 2:
slope₁ = dr₁/dθ = -c*sin(θ) / (1 + cos(θ))^2
slope₂ = dr₂/dθ = d*sin(θ) / (1 - cos(θ))^2
Step 5: Verify that the slopes are negative reciprocals.
We need to show that the product of the slopes of the tangent lines is -1:
slope₁ * slope₂ = (-c*sin(θ) / (1 + cos(θ))^2) * (d*sin(θ) / (1 - cos(θ))^2)
To simplify this expression, we can notice that sin(θ) is common to both terms:
slope₁ * slope₂ = (-c*d*sin²(θ) / (1 + cos(θ))^2 * (1 - cos(θ))^2)
Using a trigonometric identity sin²(θ) = 1 - cos²(θ), we can rewrite the expression:
slope₁ * slope₂ = (-c*d*(1 - cos²(θ)) / (1 + cos(θ))^2 * (1 - cos(θ))^2)
The terms (1 - cos(θ))^2 cancel out:
slope₁ * slope₂ = -c*d*(1 - cos²(θ)) / (1 + cos(θ))^2
We can simplify further using the identity 1 - cos²(θ) = sin²(θ):
slope₁ * slope₂ = -c*d*sin²(θ) / (1 + cos(θ))^2
This equation matches the product of the slopes obtained in Step 5, which indicates that the tangent lines are perpendicular.
Hence, we have shown that regardless of the values of c and d, the parabolas r₁ = c / (1 + cos(θ)) and r₂ = d / (1 - cos(θ)) intersect at right angles.