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AIEEE 2002

JEE Main 2002 Mathematics question on Circles

If the tangent at the point PP on the circle x2+y2+6x+6y=2x^2 + y^2 + 6x + 6y = 2 meets the straight line 5x−2y+6=05x - 2y + 6 = 0 at a point QQ on the YY-axis, then the length of PQPQ is
  1. A.
    4
  2. B.
    252\sqrt{5}
  3. C.
    5
  4. D.
    353\sqrt{5}
Show answer and solution

Correct option: C

Solution

Q lies on the Y-axis, so Q = (0, y). Substituting into 5x - 2y + 6 = 0: -2y + 6 = 0, so y = 3. Thus Q = (0, 3).

The circle is x² + y² + 6x + 6y - 2 = 0, center C = (-3, -3), radius r = √(9 + 9 + 2) = √20 = 2√5.

Since PQ is tangent to the circle from external point Q, the length of the tangent from Q to the circle = √(0² + 3² + 6(0) + 6(3) - 2) = √(9 + 18 - 2) = √25 = 5.

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