find the locus

Question

locus of the point of intersection of tangents to the circle x² + y² + 2x + 4y - 1 = 0 which include an angle of 60⁰ is

(I)x² + y² + 2x + 4y - 19 = 0

 

(ii)x² + y² + 2x + 4y + 19 = 0

 

(iii)x² + y² - 2x - 4y - 19 = 0

 

(iv)x² + y² - 2x - 4y +19 = 0

Answer 1

If you're talking about finding the equation of the circle that passes through where these tangents intersect, don't think too much about the equation at first. 

 

Draw a circle, preferably on some graph paper or something. Draw a tangent line to it somewhere. Notice that if you do this and you try to draw another one that's tangent to the circle and makes a 60 degree angle with it, that you'll be drawing one vertex of an equilateral triangle that encloses that circle. 

 

Go ahead and draw this other tangent.

 

Draw lines from the center of the circle to those points where the tangent lines hit the circle. These two new lines will make an angle of 120 degrees. 

 

Now draw a new line from the center of the circle to the point where the tangents intersect. This will bisect that 120 degree angle. 

 

Note that tangent lines always make a 90 degree angle with the radius of the circle. 

 

So you're now looking at two right triangles pressed up against each other along their hypotenuses. The length of this is the radius of the circle you're looking to find.

 

So you have a 30/60/90 right triangle. You know that the radius of the circle you started with is the shortest leg and that the radius of the circle you're trying to find is also the hypotenuse of this same triangle, which has to be double its shortest leg.

 

So the radius of your new circle is twice the radius of the circle you started with.

 

To find the equation you're looking for, make sure that it is written so it's centered at the same point.

 

 Complete the square on the equation you're given to find the center. 

 

x² + 2x + __ - __ + y² + 4y + __ - __ = 1

 

Once you have the equation in this form, you can tell what the equation of the circle centered at the same point with a radius twice what this one's is will look like. Use this to find the correct answer.

 

Message back if you need any more help.

 

Right away you're going to eliminate everything but (i) or (ii) because you'll get a different center if you complete the square on the others. If you start to complete the square on (ii) you'll notice that you'll end up with two square terms equal to a negative number, so that won't do.