Given that a circle has an equation of x^2+y^2+2x-6y-6=0, what is the exact length of AB given that a tangent is drawn from A(6,5) and touches the circle at B?

Question

Genesis Allen · Accepted Answer

#abs(AB)=sqrt(37)#

Rewriting #x^2+y^2+2x-6y-6=0#
into standard form for a circle:

#color(white)("XXX")x^2+2xcolor(orange)(+1)=y^2-6ycolor(orange)(+9) = 6 color(orange)(+1+9)#

#color(white)("XXX")(x+1)^2+(y-3)^2=4^2#
which is the equation for a circle with center #C# at #(-1,3)#
and radius #r=4#

Remember that the tangent to a circle forms a right angle with the radius of the circle to the tangent point.

#abs(AC)=sqrt((6-(-1))^2+(5-3)^2)=sqrt(49+4)=sqrt(53)#

#abs(BC)= "the radius" = 4#

#abs(AB)^2= abs(AC)^2-abs(BC)^2#

#color(white)("XXX")=53-16 = 37#

#abs(AB)=sqrt(37)#

Eva Abramson · Answer

undefined To find the length of AB, we need to first find the coordinates of the point of tangency, B, where the tangent line touches the circle. Then, we can calculate the distance between points A and B using the distance formula.

To find the coordinates of point B, we'll first complete the square for the given circle equation to put it in standard form. Then, we can determine the center and radius of the circle.

After that, we'll use the point-slope form of the equation of a line to find the equation of the tangent line passing through point A(6,5). Once we have the equation of the tangent line, we can find the point of tangency, B, by solving the system of equations formed by the tangent line and the circle equation.

Finally, we'll calculate the distance between points A and B using the distance formula:

$$ AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} $$

We'll substitute the coordinates of points A and B into this formula to find the exact length of AB.