Question

For some positive integers t, y + 5 = (1/t) * (x^2) intersects x^2 + y^2 = 25 at exactly 3 points P, Q, and R (distinct from one another). Determine the positive integers t for which the area of triangle PQR is an integer.

Answer #1

1) Let a = 〈4,−5,−2〉 and b = 〈2,−4,−5〉 Find the projection of b
onto a. proj a b=
2) Find the area of a triangle PQR, where P=(−2,−4,0),
Q=(1,2,−1), and R=(−3,−6,5)
3) Complete the parametric equations of the line through the
points (7,6,-1) and (-4,4,8)
x(t)=7−11
y(t)=
z(t)=

Determine whether the line (x,y,z) = r(t) = (1-t, 4-5t,
2t+5)
a. Intersects the xy plane
b. Intersects with the z-axis

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

6) please show steps and explanation.
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the area of the triangle
PQR.

Find the area of the triangle PQR when P = (-1, 3, 1), Q = (0,
5, 2), R = (4, 3, -1).

For what values of c is there a straight line that intersects
the curve y=x^4+cx^3+12x^2-2x+8 in 4 distinct points? answer in
interval notation

Graph y = x3 and y = x on [0, 2].
1) graph of the area between the curves y = x3 and y
= x on [0, 2] includes _____________
distinct areas.
2) One of my shaded areas has area =
3) One of my shaded areas has area =
4)
The graph showing the area between the two curves y =
x3 and y = x has four intersection points. Which of the
following points is not an intersection...

Five consecutive positive integers , p, q, r,s,t, each less than
10,000, produce a sum which is a perfect square, while the sum q+ r
+ s is a perfect cube. What is the value of the square root of
p+q+r+s+t ?

Find the point at which the line intersects the given plane.
x = -t+8 , y= 2t-2, z=6 Plane: x-4y=-6

7. For the parametric curve x(t) = 2 − 5 cos(t), y(t) = 1 + 3
sin(t), t ∈ [0, 2π) Part a: (2 points) Give an equation relating x
and y that represents the curve. Part b: (4 points) Find the slope
of the tangent line to the curve when t = π 6 . Part c: (4 points)
State the points (x, y) where the tangent line is horizontal

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