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प्रश्न
If the line y = mx does not intersect the circle (x + 10)2 + (y + 10)2 = 180, then write the set of values taken by m.
उत्तर
Let us put y = mx in the equation (x + 10)2 + (y + 10)2 = 180.
Now, we have:
(x + 10)2 + (mx + 10)2 = 180
On simplifying, we get:
\[x^2 \left( m^2 + 1 \right) + 20x\left( m + 1 \right) + 20 = 0\]
∴ Discriminant (D) = \[\sqrt{400 \left( m + 1 \right)^2 - 80( m^2 + 1)} = 4\sqrt{10}\sqrt{\left( 2m + 1 \right)\left( m + 2 \right)}\] It is given that the line y = mx does not intersect the circle (x + 10)2 + (y + 10)2 = 180.
∴ D < 0
\[\Rightarrow 4\sqrt{10}\sqrt{\left( 2m + 1 \right)\left( m + 2 \right)} < 0\]
\[ \Rightarrow m \in \left( - 2, \frac{- 1}{2} \right)\]
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