Simplify the equation to solve for k. Since the points are distinct, k cannot be 0. Therefore, k = (11)/(5). Finally, substitute the values of k and r² into the expression 10k + r² to get the final answer.

Simplify the equation to solve for k. Since the points are distinct, k cannot be 0. Therefore, k = (11)/(5). Finally, substitute the values of k and r² into the expression 10k + r² to get the final answer.

If the four distinct points $(4{,}6)$ , $(-1{,}5)$ , $(0{,}0)$ and $(k,3k)$ lie on a circle of radius $r$ , then $10k + r^2$ is equal to:

Answer: 35

Step-by-step solution

Step 1: General Equation of a Circle

The general equation of a circle is given by $x^2 + y^2 + 2gx + 2fy + c = 0$ . 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 $\sqrt{g^2 + f^2 - c}$ .

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 $r = \sqrt{g^2 + f^2 - c}$ .

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, $k = \frac{11}{5}$ . Finally, substitute the values of $k$ and $r^2$ into the expression $10k + r^2$ to get the final answer.

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If the four distinct points (4,6)(4{,}6)(4,6), (−1,5)(-1{,}5)(−1,5), (0,0)(0{,}0)(0,0) and (k,3k)(k,3k)(k,3k) lie on a circle of radius rrr, then 10k+r210k + r^210k+r2 is equal to: