Question
Match the conic section with the equation in the column on the right that represents that type of conic section._____ A parabola opening to the right or to the lefta) $ rac{x^{2}}{10}+ rac{y^{2}}{12}=1$b) $(x+1)^{2}+(y-3)^{2}=30$c) $y-x^{2}=5$d) $ rac{x^{2}}{9}- rac{y^{2}}{10}=1$e) $x-2 y^{2}=3$f) $ rac{y^{2}}{20}- rac{x^{2}}{35}=1$g) $3 x^{2}+3 y^{2}=75$h) $ rac{(x-1)^{2}}{10}+ rac{(y-4)^{2}}{8}=1$
Match the conic section with the equation in the column on the right that represents that type of conic section. _____ A parabola opening to the right or to the left a) $\frac{x^{2}}{10}+\frac{y^{2}}{12}=1$ b) $(x+1)^{2}+(y-3)^{2}=30$ c) $y-x^{2}=5$ d) $\frac{x^{2}}{9}-\frac{y^{2}}{10}=1$ e) $x-2 y^{2}=3$ f) $\frac{y^{2}}{20}-\frac{x^{2}}{35}=1$ g) $3 x^{2}+3 y^{2}=75$ h) $\frac{(x-1)^{2}}{10}+\frac{(y-4)^{2}}{8}=1$
Answers
Match the conic section with the equation in the column on the right that represents that type of conic section.
_____ A parabola opening to the right or to the left
a) $\frac{x^{2}}{10}+\frac{y^{2}}{12}=1$ b) $(x+1)^{2}+(y-3)^{2}=30$ c) $y-x^{2}=5$ d) $\frac{x^{2}}{9}-\frac{y^{2}}{10}=1$ e) $x-2 y^{2}=3$ f) $\frac{y^{2}}{20}-\frac{x^{2}}{35}=1$ g) $3 x^{2}+3 y^{2}=75$ h) $\frac{(x-1)^{2}}{10}+\frac{(y-4)^{2}}{8}=1$
In this problem, I have to use the equation that the general equation to decide what sort of shape I'm dealing with and also to put it into standard form. So knowing that I have a Y square term but no X Square term, that's a giveaway, that this is going to be a parabola. And since it's a Y Square, that means it's gonna be in this family of paralysis. So it's either going to go to the left or to the right. It will be more complicated than this basic parent function, but I know it's gonna be in the family where Ko's either left or right. So I'm gonna go ahead and put my Y terms together. So I have Why squared minus for why? And then knowing that I'm gonna want to move my ex terms over to the other side, I'll just go ahead and add the two X over here. So I'm looking at why squared minus four y equals two x minus 10. Then I know that I need to complete the square on the left. So I'm gonna look at this. Be term, which is negative, for he is an X squared plus BX plus e. I'm gonna divide it by two. So that is negative too. And then I'm gonna take that value and square it. And that's what I'll add to both sides because I can't just randomly add to one side. I have to always keep the balance of the equation. So since I have, why squared minus four y plus four. And I know I want this factor. Don't just go ahead and say the quantity Why minus two square equals two X. And since I have negative 10 plus four, I will make this a negative six. And I'm pretty close to the format I want. But it's gonna look like this when I'm all done. So I need to make some slight adjustments by factoring on the right so I'll go ahead and do that I have. Why minus two squared equals two x minus three. And what this tells me is my Vertex is always a TSH K and you can see in the K spot I have this too. And in the H spot I have this threesome, I vertex will be 32 I also know that the A value is too so Here's my equation I'm gonna keep Here's my vertex I'm gonna need for when I graphed And since a equals two and a always equals one over four C then I know See, in this particular equation is gonna end up being one ace, and that's gonna help me draw focus and give the width of the graph. So I went ahead and graft this one already for you, and you're gonna want to see it right here. So I've got my Vertex at 32 And then the shape of this is being determined by the focus, which is sea away on. In this case, it's 1/8 and again it's facing to the right as we anticipated. So here is the graph of this original problem. Why Squared minus two x minus four y equals negative 10 which in standard form became the quantity y minus two squared equals two times the quantity x minus three
Okay, so we want to find the equation or the hyper below with a horizontal access. And so horizontal hyper Poulos have the X squared over a squared minus y squared over B squared or as the vertical axis hyperba less have the Y squared over a squared minus X squared over B squared. So for this one, we can look to answer choice D, which is has the X squared first minus y squared. So we know that's the right answer, since we want the horizontal access and we have X squared first in both of these.
Okay. So for this one, we want a high purple up with a vertical axis. And so, because the horizontal hyper Poulos X squared minus y squared, we want vertical one, which is why squared minus X squared. So that answer would be asked because both of these have Why squared? Why squared minus X squared.
Okay, So for this one, we want the equation of a no lips not centered at the origin. So here in black, we have thought basic form for the equation of the lips and x one and why one represent the center. So her on the lips centered at the origin X one and why one would be zero, because the origin is 00 So for this one, the answer would be beach, because here we have X one for one. And why one just for as, um the center, not being 00 Um and sometimes also instead of X one y one, you would see HK to represent the center, but they're the same.