The equation of the red ellipse in the figure shown is x^2/25 + y...

Question

The equation of the red ellipse in the figure shown is x^2/25 + y^2/9 =1Write the equation for each circle shown in the figure.

Graph showing a red ellipse and three circles in a coordinate plane.

Accepted Answer

Welcome back. I am so glad you're here. We are told the blue ellipse shown in the figure is defined by the equation X squared divided by plus Y squared, divided by 81 equals one. We're asked to determine the equation of the two circles. We are given a figure which is a graph containing three shapes. And the first shape we're told about is the blue ellipse. The blue ellipse is centered about the origin and along the X axis, the blue ellipse intersects with a small green circle which is also centered about the origin along the Y axis. The blue ellipse intersects with a larger red circle which is also centered about the origin. For our answer choices. We are given answer choice. A, the smaller circle has an equation of X squared plus Y squared equals 36. The larger circle has an equation X squared plus Y squared equals answer. Choice B the smaller circle has an equation X squared plus Y squared equals six. The larger circle has an equation X squared plus Y squared equals nine answer choice C the smaller circle has an equation X squared plus Y squared equals three. And the larger circle has an equation X squared plus Y squared equals 4.5. And answer choice D the smaller circle has an equation X squared plus Y squared equals 72. And the larger circle has an equation X squared plus Y squared equals 181. All right. So we're given the equation of that blue ellipse. And we recall from previous lessons that the equation of an ellipse will be given to us with either X squared divided by A squared plus Y squared, divided by B squared equals one or X squared divided by B squared plus Y squared, divided by A squared equals one. So which one do we have? Well, we also recall from previous lessons that A squared is going to be larger than B squared where two A represents the major axis and two B represents the minor axis. So which one is larger? Well, in our equation, 81 is larger than 36. So 81 is going to be A squared. And so we're dealing with this second equation where B squared is beneath X squared in the fraction and A squared is beneath Y squared in the fraction. So that means we could rewrite this as equals B squared and 81 equals A squared. Well, not rewriting it, but deriving those two formulas from it or equations, excuse me. So we have those two equations and from that we can solve for A and B, we can take the square root of both sides. So the square, it of 36 is going to be plus or minus six. But we only need the positive because we're dealing with distances here. So six is B and taking the square root of both sides of 81 equals A squared, that's going to be nine equals A. So what did we just learn? Well, if two A is the major axis, then a is going to be the distance from the origin to either end of the major axis. So from the origin, the major axis is the longer one, that's the y axis here. So from the origin, we can go up to the place where we encounter our ellipse and label that as nine because that's a and interestingly enough that is the distance from the origin to the larger red circle or in other words, that's the radius of our larger circle. So the radius of our larger circle is nine. Now, very similarly, we can start at the origin. And for our minor axis, the distance from the origin to our ellipse going either direction, we can go from the origin to the right until we encounter our ellipse along the x axis. That's a distance of six. And that is the radius of our smaller circle which encounters our lips along the x axis. So the radius of our smaller circle is six and what does that mean? Well, we recall from previous lessons that the formula for the or excuse me, the equation of a circle is the quantity of X minus H squared plus the quantity of Y minus K squared equals R squared where H K is the center and R is the radius. And in this case, because everything's centered about the origin, the center is going to be 00 for both of these circles. And now we can fill in for H K and R to get our equations. So for the larger circle, we have X minus zero squared plus Y minus zero squared equals nine squared while X minus zero is X. So that's just X squared plus Y minus zero is Y. So that's just Y squared and that equals nine squared is 81. Now, for our smaller circle filling that in, we have X minus again, zero squared plus Y minus K oops, not K zero squared, not equals R here is six squared. And again, this is just X squared plus Y squared equals and six squared is 36. And those are our equations for our circles. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

The equation of the red ellipse in the figure shown is x^2/25 + y^2/9 =1Write the equation for each circle shown in the figure.

Key Concepts

Ellipse Equation

Circle Equation

Graphing Conic Sections

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