# Given #L_1->x+3y=0#, #L_2=3x+y+8=0# and #C_1=x^2+y^2-10x-6y+30=0#, determine #C->(x-x_0)^2+(y-y_0)^2-r^2=0# tangent to #L_1,L_2# and #C_1#?

See below.

Firstly we will pass the geometrical objects to a more convenient representation.

Here

Now, given

and

with

If

with

Other conditions for

but

so

and

The essential set of equations to obtain the solution is

two equations and two incognitas

Solving for

The attached plot shows the answer in red and the initial elements in black.

By signing up, you agree to our Terms of Service and Privacy Policy

To find the circle tangent to the lines ( L_1: x + 3y = 0 ), ( L_2: 3x + y + 8 = 0 ), and the circle ( C_1: x^2 + y^2 - 10x - 6y + 30 = 0 ), we need to follow these steps:

- Find the point of intersection of ( L_1 ) and ( L_2 ) to determine the center of the circle.
- Calculate the distance between the center of the circle and the given circle ( C_1 ) to determine the radius of the circle.

First, let's find the point of intersection of ( L_1 ) and ( L_2 ):

Solving the system of equations formed by ( L_1 ) and ( L_2 ), we get the point of intersection ( (x_0, y_0) ).

Next, calculate the distance between ( (x_0, y_0) ) and the center of the circle ( C_1 ) using the formula for the distance between two points:

[ d = \sqrt{(x_0 - x_1)^2 + (y_0 - y_1)^2} ]

where ( (x_1, y_1) ) is the center of ( C_1 ).

This distance represents the radius of the circle ( C ). Therefore, the equation of the circle tangent to ( L_1 ), ( L_2 ), and ( C_1 ) is:

[ (x - x_0)^2 + (y - y_0)^2 - r^2 = 0 ]

where ( (x_0, y_0) ) is the center of the circle and ( r ) is the radius of the circle.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Circle A has a center at #(5 ,3 )# and a radius of #1 #. Circle B has a center at #(0 ,-5 )# and a radius of #2 #. Do the circles overlap? If not, what is the smallest distance between them?
- Circle A has a center at #(-8 ,8 )# and a radius of #8 #. Circle B has a center at #(-3 ,3 )# and a radius of #4 #. Do the circles overlap? If not, what is the smallest distance between them?
- How do I find the equation of the perpendicular bisector of the line segment whose endpoints are (-4, 8) and (-6, -2) using the Midpoint Formula?
- What is the perimeter of a triangle with corners at #(9 ,2 )#, #(6 ,3 )#, and #(4 ,7 )#?
- A triangle has corners at #(1 ,9 )#, #(5 ,4 )#, and #(6 ,8 )#. How far is the triangle's centroid from the origin?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7