Question

The range of values of a for which the point (a, 4) is outside the circles $x ^ { 2 } + y ^ { 2 } + 10 x = 0$ and $x ^ { 2 } + y ^ { 2 } - 12 x + 20 = 0$ is

• A. $( - \infty , - 8 ) \cup ( - 2,6 ) \cup ( 6 , + \infty )$
• B. (– 8, – 2)
• C. $( - \infty , - 8 ) \cup ( - 2 , + \infty )$
• D. None of these

Answer 1

Answer: (A)

$( - \infty , - 8 ) \cup ( - 2,6 ) \cup ( 6 , + \infty )$

Answer 2

For circle, $x ^ { 2 } + y ^ { 2 } + 10 x = 0$;

$a ^ { 2 } + ( 4 ) ^ { 2 } + 10 a > 0 \Rightarrow a ^ { 2 } + 10 a + 16 > 0$ $\Rightarrow \quad ( a + 8 ) ( a + 2 ) > 0$

$\Rightarrow a < - 8$ or $a > - 2$ …..(i)

For circle, $x ^ { 2 } + y ^ { 2 } - 12 x + 20 = 0$ ;

$a ^ { 2 } + ( 4 ) ^ { 2 } - 12 a + 20 > 0 \Rightarrow a ^ { 2 } - 12 a + 36 > 0$

$\Rightarrow \quad a \in R$∼$\{ 6 \}$…..(ii)

Taking common values from (i) and (ii),

$a \in ( - \infty , - 8 ) \cup ( - 2,6 ) \cup ( 6 , + \infty )$