Did you get a chance to try the tricky geometry problem from Monday?
ACT Challenge Question #7
Which of the following is an equation of the largest circle that can be inscribed in the ellipse with the equation?
Answer and Explanation
This question is difficult primarily because you need to have some higher-level knowledge of the equations of circles and ellipses. If you just studied these in school, then it wouldn’t be nearly as tough. But because equations of ellipses and circles are fair game for higher-level questions on the ACT, we thought we’d feature one here in our Challenge series!
So let’s get started with the equation for an ellipse:
+ = 1
+ = 1
If an ellipse is wider than it is tall, our equation looks like the first one. If it is taller than it is wide, it looks like the second one. The a always goes with the variable whose axis parallels the wider direction of the ellipse, and the b always goes with the variable whose axis parallels the narrower direction, hence the reason for the difference in the equations.
In our problem, we have an ellipse that is taller than it is wide. What’s important is that you visualize what an ellipse like this looks like:
If we put a circle inside this shape, it can only have a diameter that is as wide as the ellipse is wide, otherwise it wouldn’t fit.
Ok back to the ellipse equation:
h and k tell us where the center of the ellipse is, so our center is (2,-4). a and b tell us how many units away from the center each vertex is. Since a is 3 and b is 4, we know that our ellipse is 6 units wide and 8 units tall.
If we put a circle in this ellipse, this means it cannot have a radius greater than 3.
The equation of a circle is:
So we need to see , or 9, in our answer choice for the radius of the circle, making our answer C.