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Circle question

2017 · 9 Apr · Shift 1 · Q44
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Circle question

2017 · 9 Apr · Shift 1 · Q44

JEE MainMathematicsCircleMCQ+4 / −1
A line drawn through the point P(4, 7) cuts the circle x2 + y2 = 9 at the points A and B. Then PA⋅PB is equal to :
  1. A
    53
  2. B
    56
  3. C
    74
  4. D
    65
View written solutionFree

Correct answer: B

  1. We use the power of a point theorem.

    For a point P(x1,y1)P(x_1,y_1)P(x1​,y1​) and a circle x2+y2=r2,x^2+y^2=r^2,x2+y2=r2, if a secant through PPP meets the circle at AAA and BBB, then PA⋅PB=OP2−r2,PA\cdot PB = OP^2-r^2,PA⋅PB=OP2−r2, where OOO is the center of the circle.

  2. Here, the circle is x2+y2=9,x^2+y^2=9,x2+y2=9, so its center is O(0,0)O(0,0)O(0,0) and radius is r=3.r=3.r=3.

  3. The given point is P(4,7)P(4,7)P(4,7). Compute OP2OP^2OP2: OP2=42+72=16+49=65.OP^2 = 4^2+7^2=16+49=65.OP2=42+72=16+49=65.

  4. Therefore, PA⋅PB=OP2−r2=65−9=56.PA\cdot PB = OP^2-r^2 = 65-9 = 56.PA⋅PB=OP2−r2=65−9=56.

  5. So the correct option is 56.\boxed{56}.56​.

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