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Math Help - Some Hyperbola, ellipse, parabola help.

  1. #1
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    Some Hyperbola, ellipse, parabola help.

    I have 13 questions out of a larger group that I need some direction with.
    If anyone wants to answer them, go for it, if not get me started please. I did fine with my others, these I can only think of bits or pieces I may need.
    Thanks ahead of time.

    1. Determine the perpendicular distance between the parallel lines y=2x+3 and y=2x-5.
    2. Find the sum of the x and y intercepts of the line x/4 + y/8=2.
    3. Find the focal width of the parabola y=2(x)squared.
    4. Find the focus of the parabola ysquared-6y-4x=17=0.
    5. Which of the conic sections have the property that the ratio of the distance from a point P on the curve to a fixed point F to the distance from P to a fixed line is a positive constant less than 1?
    6. A line is drawn tangent to the ellipse xsquared + 3ysquared = 84 at the point (3,5). Determine the area in the first quadrant below the tangent line.
    7. Find the foci of the hyperbola xsquared/17 - ysquared/8 = 1.
    8. Determine the eccentricity of the hyperbola 2ysqrd-3xsqrd=1.
    9. Determine the asymptotes of the hyperbola xsqrd-ysqrd/4=1.
    10. Find an equation for the hyperbola with vertices (0,+/- 3) and foci (0,+/- 5).
    11. Find the vertices of the hyperbola 9ysqrd-18y-4xsqrd-16x-43=0.
    12. Find the equations for the directrices of the hyperbola xsqrd/16-ysqrd/9=1.
    13. Find an equation for the ellipse with vertices (+/- 3,0) and directrices x=+/-4.
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  2. #2
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    Quote Originally Posted by dmbocci View Post
    I have 13 questions out of a larger group that I need some direction with.
    If anyone wants to answer them, go for it, if not get me started please. I did fine with my others, these I can only think of bits or pieces I may need.
    Thanks ahead of time.

    1. Determine the perpendicular distance between the parallel lines y=2x+3 and y=2x-5.
    2. Find the sum of the x and y intercepts of the line x/4 + y/8=2.
    ...
    I haven't much time yet so I'll help you with the first 2 questions:

    1. If you have the equation of a line in the form

    Ax +By +C=0 then d=\dfrac{C}{\sqrt{A^2+B^2}} yields the distance of the origin to this line.
    Since your lines are parallel you only have to calculate the difference of the 2 distances of the origin.

    Therefore d_1 = \dfrac{3}{\sqrt{5}} and d_2 = \dfrac{-5}{\sqrt{5}} which will give \dfrac{8}{\sqrt{5}}\approx 3.5777

    2. The equation \dfrac xa+\dfrac yb = 1 describes a line with the x-intercept a and the y-intercept b.

    \dfrac x4 + \dfrac y8=2~\implies~\dfrac x8 + \dfrac y{16}=1

    Thus the sum of the intercepts is 24.
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  3. #3
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    Quote Originally Posted by dmbocci View Post
    ...
    3. Find the focal width of the parabola y=2(x)squared.
    4. Find the focus of the parabola ysquared-6y-4x=17=0.
    ...
    to #3:

    The general equation is x^2=4py where p is the distance between the vertex and the focus of the parabola.

    y=2x^2~\implies~x^2=\dfrac12 y~\implies~x^2=4\cdot \dfrac18 y

    Since the vertex is at V(0, 0) the focus is at F\left(0, \frac18\right)

    The focal width is 2p = \frac14


    to #4:

    I assume that you mean:

    y^2-6y-4x+17=0~\implies~y^2-6y\bold{\color{red}+9} = 4x-17\bold{\color{red}+9} ~\implies~(y-3)^2=4\cdot 1\cdot(x-2)

    Therefore the vertex is at V(2, 3), p = 1 thus F(2+1,3) thus F(3, 3)
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