CircleHardBloom L3
Question
The equation of a line inclined at an angle $\frac{\pi}{4}$ to the axis $X$, such that the two circles $x^{2} + y^{2} = 4,x^{2} + y^{2} - 10x - 14y + 65 = 0$ intercept equal lengths on it, is:
Options
A.$2x - 2y - 3 = 0$
B.$2x - 2y + 3 = 0$
C.$x - y + 6 = 0$
D.$x - y - 6 = 0$
Solution
[Key Concept: A $45^{\circ}$ line related to two circles: impose the tangency/equal-chord condition.]
$r_{1}^{2} - P_{1}^{2} = r_{2}^{2} - P_{2}^{2}$
$${\Rightarrow 4 - \frac{C^{2}}{2} = 9 - \frac{(2 - C)^{2}}{2} }{\Rightarrow C = - \frac{3}{2} }$$⇒ equation of line $y = x - \frac{3}{2}$
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