JEE MAIN - Mathematics (2010 - No. 16)
The circle $${x^2} + {y^2} = 4x + 8y + 5$$ intersects the line $$3x - 4y = m$$ at two distinct points if :
$$ - 35 < m < 15$$
$$ 15 < m < 65$$
$$ 35 < m < 85$$
$$ - 85 < m < -35$$
Explanation
Circle $${x^2} + {y^2} - 4x - 8y - 5 = 0$$
Center $$=(2,4),$$ Radius $$ = \sqrt {4 + 16 + 5} = 5$$
If circle is intersecting line $$3x-4y=m,$$ at two distinct points.
$$ \Rightarrow $$ length of perpendicular from center to the line $$ < $$ radius
$$ \Rightarrow {{\left| {6 - 16 - m} \right|} \over 5} < 5 \Rightarrow \left| {10 + m} \right| < 25$$
$$ \Rightarrow - 25 < m + 10 < 25 \Rightarrow - 35 < m < 15$$
Center $$=(2,4),$$ Radius $$ = \sqrt {4 + 16 + 5} = 5$$
If circle is intersecting line $$3x-4y=m,$$ at two distinct points.
$$ \Rightarrow $$ length of perpendicular from center to the line $$ < $$ radius
$$ \Rightarrow {{\left| {6 - 16 - m} \right|} \over 5} < 5 \Rightarrow \left| {10 + m} \right| < 25$$
$$ \Rightarrow - 25 < m + 10 < 25 \Rightarrow - 35 < m < 15$$
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