A line y = m x + 1 intersects the circle x - 3 2 + y + 2 2 = 25 at the points P and Q . If the midpoint of the line segment P Q has x - coordinate - 3 5 , then which one of the following options is correct?

Step-by-step solution

Equation of line P Q : y = m x + 1 Let R be the mid point of P Q , so its x -coordinate of midpoint = - 3 5 Hence, coordinates of midpoint R - 3 5 , - 3 m 5 + 1 Let centre of circle is O 3 , 2 Since, perpendicular dropped from centre upon any chord divides it in two equal parts. Hence, line P Q will be perpendicular to R O ⇒ m R Q ⋅ m R O = - 1 ⇒ m - 3 m 5 + 1 + 2 - 3 5 - 3 = - 1 ⇒ m - 3 m + 15 - 18 = - 1 ⇒ m 2 - 5 m + 6 = 0 ⇒ m = 2 , 3 ⇒ 2 ≤ m 4