In the standard x,y plane, a circle has a radius 6 and center (7, 3). The circle intersects the x-axis at (a, 0) and (b, 0). What is the value of a + b?
Accepted Solution
A:
Answer: The required value of a+b is 14.Step-by-step explanation: Given that in the standard xy plane, a circle has a radius 6 and center (7, 3).The circle intersects the x-axis at (a, 0) and (b, 0). We are to find the value of a+b.We know thatthe standard equation of a circle with center at (h, k) and radius r units is given by[tex](x-h)^2+(y-k)^2=r^2~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]For the given circle, we have(h, k) = (7, 3) and r = 6 units.So, from equation (i), we get[tex](x-7)^2+(y-3)^2=6^2\\\\\Rightarrow (x-7)^2+(y-3)^2=36~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]Since the circle (ii) passes through the points (a, 0) and (b, 0), so let the point be denoted by (c, 0), then we have[tex](c-7)^2+(0-3)^2=36\\\\\Rightarrow (c-7)^2+9=36\\\\\Rightarrow (c-7)^2=27\\\\\Rightarrow c-7=\pm3\sqrt3~~~~~~~~~~~~~~~~~~~~~~[\textup{Taking square root on both sides}]\\\\\Rightarrow c=7\pm3\sqrt3[/tex]Therefore, we get[tex]a=7+3\sqrt3,\\\\b=7-3\sqrt3.[/tex]That is,[tex]a+b=(7+3\sqrt3)+(7-3\sqrt3)=14.[/tex]Thus, the required value of a+b is 14.