Question in analytical geometry

In summary, we have found that the equation of a circle with center on the line y = -2x and containing the point (1, 0) can be expressed as (x - a)^2 + (y + 2a)^2 = 5a^2 - 2a + 1. By solving for the intersection point with the line 3x + 4y + 15 = 0, we can find the values of a and the resulting equations of the circles, which are (x - 1)^2 + (y + 2)^2 = 4 and (x + 2)^2 + (y - 4)^2 = 25.
  • #1
Chen
977
1
The point (1, 0) is on a circle, which center is on the line y = -2x. The line 3x + 4y + 15 = 0 is a tangent to the circle. Find the equation(s) for the circle(s).

Thanks,
 
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  • #2
The equation of a circle is (x - a)^2 + (y - b)^2 = r^2.

The centre is on the line y = -2x, which means b = -2a, hence our circle has the equation:

(x - a)^2 + (y - (-2a))^2 = r^2
<=>
(x - a)^2 + (y + 2a)^2 = r^2

Since (1, 0) was on the circle:

(1 - a)^2 + (0 + 2a)^2 = r^2
<=>
r^2 = 5a^2 - 2a + 1

Putting that back into the equation of our circle:

(x - a)^2 + (y + 2a)^2 = 5a^2 - 2a + 1 ... (1)

We know that (1) must intersect with 3x + 4y + 15 = 0 somewhere, find that point. We know that at that point, the derivative of our circle's equation with respect to x must be -3/4 (i.e the slope of the tangent line), which gives us an equation where we can solve for a.

But I'll be damned if I'm going to do all those nasty calculations by hand ;) I worked them out with my computer, which gave a = -2 and a = 1, so the equations of the circles might be these:

(x - 1)^2 + (y + 2)^2 = 4
(x + 2)^2 + (y - 4)^2 = 25

They seem to work...
 
Last edited:
  • #3
Thanks for solving this. We are actually not allowed to use things like derivative calculations in this type of questions, so I took a different approach. But the answers are the same so I'm happy. :)
 

What is analytical geometry?

Analytical geometry is a branch of mathematics that combines algebra and geometry to study geometric shapes and their properties using coordinate systems and equations.

What are the key concepts in analytical geometry?

The key concepts in analytical geometry include points, lines, curves, and surfaces in a coordinate system, as well as concepts such as slope, distance, and the equations that represent these shapes.

How is analytical geometry used in real life?

Analytical geometry is used in various fields such as engineering, physics, computer graphics, and navigation. It is used to model and analyze real-world problems, design structures and machines, and create visual representations of data.

What are some common equations used in analytical geometry?

Some common equations used in analytical geometry include the equation of a line (y=mx+b), the distance formula (d=√((x2-x1)^2+(y2-y1)^2)), and the equation of a circle ((x-h)^2+(y-k)^2=r^2).

How can I improve my understanding of analytical geometry?

To improve your understanding of analytical geometry, you can practice solving problems, work on visualizing and interpreting geometric shapes, and study the key concepts and equations. It can also be helpful to seek out additional resources such as textbooks, online tutorials, and practice exercises.

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