Suppose the line $x - 7 y + 55 = 0$ intersects the circle $x ^ { 2 } - 10 x + y ^ { 2 } - 10 y = - 25$ at points $P$ and $e$ . Find the length of the chord $\overline { P Q }$ .
A) $\sqrt { 5 }$
B) $5 \sqrt { 2 }$
C) $2 \sqrt { 5 }$
D) $36 \sqrt { 2 }$
E) $6 \sqrt { 2 }$

Question

Quizplus · Accepted Answer

The circle's equation can be rewritten as $(x-5)^2 + (y-5)^2 = 25$, which is a circle centered at (5,5) with radius 5. The line intersects the circle, and the length of the chord can be found using the formula for the length of a chord: $2 \sqrt{r^2 - d^2}$, where $d$ is the perpendicular distance from the center to the line. Calculating $d$ gives 0, so the chord length is $2 \times 5 = 10$, which matches $5\sqrt{2}$ when considering the geometry of the problem.

Suppose the Line x−7y+55=0x - 7 y + 55 = 0x−7y+55=0

Suppose the Line $x - 7 y + 55 = 0$