Question

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

• A. 32
• B. 33
• C. 34
• D. 35

Hint:

If multiple points lie on a circle, the perpendicular bisectors of the chords formed by these points are concurrent at the center of the circle. Alternatively, use the general equation of a circle and substitute the coordinates of the given points to form a system of equations to find the center and radius. If three points form a right-angled triangle with the right angle at one of the given points, then the hypotenuse is the diameter of the circle.

Answer 1

The Correct Option is D

\(\text{The points } (4, 6), (-1, 5) \text{ and } (0, 0) \text{ lie on the circle.}\)
\(\text{Let the equation of the circle be } x^2 + y^2 + 2gx + 2fy + c = 0.\)
\(\text{Since } (0, 0) \text{ lies on the circle, } c = 0.\)
\(\text{Substitute } (4, 6):\)
\(16 + 36 + 8g + 12f = 0 \Rightarrow 8g + 12f = -52 \Rightarrow 2g + 3f = -13 \quad \text{...(i)}\)
\(\text{Substitute } (-1, 5):\)
\(1 + 25 - 2g + 10f = 0 \Rightarrow -2g + 10f = -26 \Rightarrow -g + 5f = -13 \quad \text{...(ii)}\)
\(\text{Add (i) and 2 × (ii):}\)
\(2g + 3f + 2(-g + 5f) = -13 + 2(-13)\)
\(\Rightarrow 2g + 3f - 2g + 10f = -13 - 26 \Rightarrow 13f = -39 \Rightarrow f = -3\)
\(\text{Substitute } f = -3 \text{ in (ii):}\)
\(-g + 5(-3) = -13 \Rightarrow -g - 15 = -13 \Rightarrow -g = 2 \Rightarrow g = -2\)
\(\text{Equation of the circle: } x^2 + y^2 - 4x - 6y = 0\)
\(\text{Center: } (-g, -f) = (2, 3)\)
\(\text{Radius: } r = \sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-3)^2 - 0} = \sqrt{4 + 9} = \sqrt{13}\)
\(\text{Since } (k, 3k) \text{ lies on the circle:}\)
\(k^2 + (3k)^2 - 4k - 6(3k) = 0\)
\(\Rightarrow k^2 + 9k^2 - 4k - 18k = 0\)
\(\Rightarrow 10k^2 - 22k = 0\)
\(\Rightarrow 2k(5k - 11) = 0\)
\(\text{Since points are distinct, } k \ne 0 \Rightarrow 5k - 11 = 0 \Rightarrow k = \frac{11}{5}\)
\(\text{Now, } 10k + r^2 = 10 \cdot \frac{11}{5} + 13 = 22 + 13 = 35\)