# Math Help - Some Hyperbola, ellipse, parabola help.

1. ## 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.

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.

2. Originally Posted by dmbocci
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.

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.

3. Originally Posted by dmbocci
...
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)