# Question #9a3da

##### 2 Answers

#### Explanation:

As you can see this describes a circle:

graph{x^2+y^2-y-3=0 [-10, 10, -5, 5]}

The 2 tangents where the line

We can find

To get the gradients, we can differentiate both sides implicitly since both

If

Factorising:

If

If

So these are the 2 values of

Now to get the equation of the 1st tangent where

To get

The equation for the tangent

Now to get the 2nd tangent:

To get

The equation becomes:

At the intersection of the 2 tangents we can put

Now to get the

**So the (x,y) co-ordinates of the intersection are #(13/4,1/2)#**

To get the angle of intersection between the 2 tangents there is an expression you can use but I think its best to look at the geometry to see what's going on:

Here you can see that the angle of intersection is

Michael has given a fine answer using a bit of calculus. Here is a partial answer without calculus.

#### Explanation:

The circle

The center of the circle is

The points on the circle with

To find the slopes of the tangent lines at these points, use the fact that a tangent to a circle at a point is perpendicular to the radius at that point.

At

So the slope of the tangent at

See Michael's answer to get the equation of the tangent line at

At

So the slope of the tangent at

See Michael's answer to get the equation of the tangent line at

For the remainder of the solution, see Michael's answer.