Question

The circle ${x^2} + {y^2} = 4x + 8y + 5$ intersects the line $3x - 4y = m$ at two distinct points if
A. $ - 35 < m < 15$
B. $15 < m < 65$
C. $35 < m < 85$
D. $ - 85 < m < - 35$

Answer 1

Hint: According to given in the question we have to determine the two distinct point when the circle ${x^2} + {y^2} = 4x + 8y + 5$ intersects the line $3x - 4y = m$ at two distinct points. So, first of all we have to find the centre of the given circle which can be obtained with the help of the equation of the circle which is ${x^2} + {y^2} - 2ax - 2by - c = 0$ where, a and b are the radius of the circle.
Now, we have to find the radius of the circle with the help of the formula as mentioned below:

Formula used: $ \Rightarrow r = \sqrt {{a^2} + {b^2} + c} .................(A)$
Where, r is the radius and a, b are the centre of the given circle.
Now, with the help of the line of the equation which is as $3x - 4y = m$ we have to determine the point of intersection which can be determine by the help of comparing the general equation of the line as ${a_1}x + {b_1}y + c = 0$
Now, as we know that the given equation of line $3x - 4y = m$ intersects the circle at two points is the perpendicular distance of the line from centre of the circle is less than radius with the help of the formula as mentioned below:
$ \Rightarrow \left| {\dfrac{{{a_1}a - {b_1}b - m}}{{\sqrt {{a_1}^2 + {b_1}^2} }}} \right| < r..................(B)$
Where, $({a_1},{b_1})$ are the points of intersection, r is the radius and $(a,b)$is the centre of the circle. Hence, on substituting all the values in the formula (B) above, we can determine the required range.

Complete step-by-step solution:
Step 1: First of all we have to find the radius of the given circle which can be obtained with the help of the equation of the circle which is ${x^2} + {y^2} - 2ax - 2by - c = 0$ where, a and b are the centres of the circle as mentioned in the solution hint. Hence,
$ \Rightarrow (a,b) = (2,4)$
Step 2: Now, we have to determine the radius of the circle with the help of the formula (A) as mentioned in the solution hint. Hence, on substituting all the values in the formula (A),
$
   \Rightarrow r = \sqrt {{2^2} + {4^2} + 5} \\
   \Rightarrow r = \sqrt {16 + 9 + 5} \\
   \Rightarrow r = \sqrt {25} \\
   \Rightarrow r = 5
 $
Step 3: Now, with the help of the line of the equation which is as $3x - 4y = m$ we have to determine the point of intersection which can be determine by the help of comparing the general equation of the line as ${a_1}x + {b_1}y + c = 0$. Hence, points of intersection are:
$ \Rightarrow ({a_1},{b_1}) = (3,4)$
Step 4: Now, as we know that the given equation of line $3x - 4y = m$ intersects the circle at two points is the perpendicular distance of the line from centre of the circle is less than radius with the help of the formula (B) as mentioned in the solution hint. Hence, on substituting all the values in the formula (C),
 $
   \Rightarrow \left| {\dfrac{{3(2) - 4(4) - m}}{{\sqrt {{3^2} + {4^2}} }}} \right| < 5 \\
   \Rightarrow \left| {\dfrac{{6 - 16 - m}}{{\sqrt {9 + 16} }}} \right| < 5 \\
   \Rightarrow \left| {\dfrac{{ - 10 - m}}{{\sqrt {25} }}} \right| < 5 \\
   \Rightarrow - 5 < \dfrac{{ - 10 - m}}{5} < 5
 $
Now, we have to solve the expression obtained just above, by multiplying 5 in the whole expression,
$
   \Rightarrow - 25 < - 10 - m < 25 \\
   \Rightarrow - 15 < - m < 35
 $
Now, on multiplying with -1 in the both sides of the expression just obtained above,
$ \Rightarrow - 35 < m < 15$
Hence, with the help of the formula (A) and (B) as mentioned in the solution hint we have determined the required range which is $ - 35 < m < 15$.

Therefore option (A) is correct.

Note: To determine the centre and radius of the given circle we have to compare the given circle with the general form of the equation of the circle which is ${x^2} + {y^2} - 2ax - 2by - c = 0$ where, a and b are the centre and we can determine the radius of the circle with the formula $r = \sqrt {{a^2} + {b^2} + c} $
The given line $3x - 4y = m$ intersects the circle at two points if the perpendicular distance of the line from centre of the circle is less than radius.