Question

The exhaustive range of values of a such that the angle between the pair of tangents drawn from (a, a) to the circle x² + y² – 2x – 2y – 6 = 0 lies in the range $\left( \frac { \pi } { 3 } , \pi \right)$, is –

• A. (1, )
• B. (–5, – 3) Č (3, 5)
• C. (– , – 2$\sqrt { 2 }$) Č (2$\sqrt { 2 }$, )
• D. (– 3, – 1) Č (3, 5)

Answer 1

Answer: (D)

(– 3, – 1) Č (3, 5)

Answer 2

The centre of the given circle is (1, 1) and

radius = 2$\sqrt { 2 }$. Point (a, a) must lie outside the circle,

so 2a² – 4a – 6 > 0

Ž a < – 1 or a > 3 ..........(i)

Now, tan $\frac { 2 \sqrt { 2 } } { \sqrt { 2 a ^ { 2 } - 4 a - 6 } }$

Since, $\frac { \pi } { 6 }$ < $\frac { \pi } { 2 }$

\ $< 2 \sqrt { 3 }$

\ a² – 2a – 15 < 0

Ž –3 < a < 5 .........(ii)

\ a Ī (– 3, –1 ) Č (3, 5).

[ from Eqs. (i) and (ii) ]