If the four distinct points (4,6), (−1,5), (0,0) and (k,3k) lie on a circle of radius r, then 10k+r2 is equal to:
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Step-by-Step Solution
Step 1: General Equation of a Circle
The general equation of a circle is given by x2+y2+2gx+2fy+c=0. We can use the given points to form a system of equations to find the values of g, f, and c. The center of the circle is (-g, -f) and the radius r is given by g2+f2−c.
Step 2: Substitute the given points into the equation
First, substitute the point (0,0) into the general equation, which directly gives c=0. Then, substitute the points (4,6) and (−1,5) into the equation with c=0 to obtain two linear equations in terms of g and f.
Step 3: Solve for g and f
Solve the system of linear equations for g and f. Multiply equation (2) by 2 and add it to equation (1) to eliminate g and find f. Then substitute the value of f back into one of the equations to find g.
Step 4: Find the radius r
With g=−2, f=−3, and c=0, we can calculate the radius r using the formula r=g2+f2−c.
Step 5: Substitute the fourth point (k, 3k) into the circle equation
Since the fourth point (k,3k) also lies on the circle, substitute its coordinates and the values of g, f, and c into the general equation of the circle. This will allow us to solve for k.
Step 6: Solve for k and calculate the final expression
Simplify the equation to solve for k. Since the points are distinct, k cannot be 0. Therefore, k=511. Finally, substitute the values of k and r2 into the expression 10k+r2 to get the final answer.