1. ## Math circles help

question 7 in part A in attached document

How do i show it is a tangent to the circle?

what i did was found the centre of the circle to be (1,2)

then i found the gradient and usuing the princinple that perp. lines product of gradient = -1 found the other gradient to be -1/4

after plugging the gradient i got C=13/2

what do i do next?

2. ## Re: Math circles help

There are several different ways to do that.
1) Solve the equation of the line for y and replace y in the equation of the circle by that expression,, getting a quadratic equation in x. The line is tangent to the circle if and only if the quadratic equation has a double root- if it reduces to $(x- a)^2= 0$.

2) By completing the square, find the center of the circle and so find the equation, and then slope of the line from the center of the circle to (2, 6), a radius of the circle. Show that (2, 6) satisfies the equations of both line and circle and show that the line is perpendicular to the radius by showing that the product of their slopes is -1.

3) Show that x= 2, y= 6 satisfies the equations of both line and circle and show that the slope of the line is equal to the derivative of the equation of the circle at that point.

3. ## Re: Math circles help

equation of circle put in standard form (x+1)^2 +(y-2)^2 =25
Center is ?
Slope of radius from center to (2,6) is ?
Slope of given line is?
Are the lines perpendicular
Does (2,6) satisfy both equations