Find the equation of:
1. The tangent to the circle x^2 - 2x + y^2 = 15 at(1,4)
2.the normal to the circle x^2 + 4x + y^2 - 6y = 0 at (1,1)
3. The tangent to the circle with C(1,2), r= root 5 at (3,3)
4. The normal to the cirlcle x^2 - 4x + y^2 = 9 at (5,2)
5. Show that the point P(-3,2) lies on the circle (x-1)^2 + (y+2)^2 = 32. Find the tangent to the circle at P.
6. The line 3x - y = 3 is tangent to the circle with centre (5,-1) and radius r. Find the value of r
1. The tangent to the circle x^2 - 2x + y^2 = 15 at(1,4)
Implicitly differentiating the function, we can find the slope of a tangent line to the circle
2x - 2 + 2yy' = 0
y' = [2 - 2x ] / [2y]
y' = [1 - x] / y
At (1,4)....the slope is
[1 - 1 ] / 4 = 0
So...the equation of the tangent line at (1,4) is
y = 4
Here's the graph : https://www.desmos.com/calculator/uqvstutnla
2.the normal to the circle x^2 + 4x + y^2 - 6y = 0 at (1,1)
The slope of a tangent line at any point is found as
2x + 4 + 2yy' - 6y' = 0
y' [ 2y - 6] = - [2x + 4]
y ' = - [2x + 4 ] / [2y - 6]
At (1,1), the slope is
-[2(1) + 4 ] / [ 2(1) - 6 ] = - 6 / -4 = 3/2
Since we want a normal line at this point...the slope of our line is -2/3
So...the equation of the normal line is
y = (-2/3) ( x - 1) + 1
y =(-2/3)x + 2/3 + 1
Here's the graph : https://www.desmos.com/calculator/1jrzfuhcxk
3. The tangent to the circle with C(1,2), r= root 5 at (3,3)
The slope between the center and the point is
[ 3 - 2] / [ 3 - 1] = 1/2
The tangent line will have a negative reciprocal slope = -2
So.....the equation of the tangent line is
y = -2 ( x -3) + 3
y = -2x + 6 + 3
y = -2x + 9
Here's the graph : https://www.desmos.com/calculator/ve37r2kubn
4. The normal to the cirlcle x^2 - 4x + y^2 = 9 at (5,2)
Let's complete the square to find the center
x^2 -4x + 4 + y^2 = 4 + 9
(x - 2)^2 + (y - 0)^2 = 13
The center is (2, 0)
The slope between this point and ( 5,2) is
[ 2- 0 ] / [ 5 - 2 ] = 2/3 = slope of the normal line
The equation of the normal line at (5,2) is
y = (2/3) (x - 5) + 2
y = (2/3)x - 10/3 + 2
y= (2/3)x - 4/3
Here's the graph : https://www.desmos.com/calculator/u8n7sc01uh