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Question:

The circle x² + y² = 4x + 8y + 5 intersects the line 3x - 4y = m at two distinct points if?

35 < m < 85

-35 < m < 15

-85 < m < -35

15 < m < 65

Solution:

Circle x² + y² - 4x - 8y - 5 = 0
Centre = (2, 4), Radius = √(4 + 16 + 5) = 5
If circle is intersecting line 3x - 4y = m at two distinct points.
Length of perpendicular from center < radius
=|6 - 16 - m|/5 < 5
=|10 + m| < 25
-25 < m + 10 < 25
-35 < m < 15