Multiple choice

Consider $L_{1}: 2x + 3y +p -3 = 0 L_{2}: 2x + 3y + p + 3 = 0,$ where p is a real number, and $C:x^{2}+y^{2}+6x-10y+ 30= 0$ Statement-1: If line $L_{1}$ is a chord of circle $C$, then $L_{2}$ is not always a diameter of circle $C.$

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is correct but Reason is incorrect

  4. Both Assertion and Reason are incorrect
Reveal answer Fill a bubble to check yourself
C Correct answer
AI explanation

The circle x^2 + y^2 + 6x - 10y + 30 = 0 has center (-3, 5) and radius 4. Using the perpendicular distance from the center to L1 (2x + 3y + p - 3 = 0), we get |-6 + 15 + p - 3| / sqrt(13) = |p + 6| / sqrt(13); for L1 to be a chord, this distance must be strictly less than 4, so -4*sqrt(13) - 6 < p < 4*sqrt(13) - 6. The distance between the parallel lines L1 and L2 is |-3 - 3| / sqrt(13) = 6 / sqrt(13), which is less than the diameter of 8, confirming L2 intersects the circle and is thus not always a diameter.

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