Let L be a common tangent line to the curves $$4x2$$ $$+$$ $$9y2$$ $$=$$ $$36$$ and $$(2x)2$$ $$+$$ $$(2y)2$$ $$=$$ $$31$$. Then the square of the slope of the line L $$is_____________$$

Given : Ellipse is $$x29+y24=1let$$ tangent be $$y=mx$$±$$9m2+4$$ It is tangent to the circle $$x2+y2=314then$$ ⇒$$9m2+4m2+1=312$$ ⇒$$4(9m2+4)=31(m2+1)$$ ⇒$$36m2-31m2=31-16$$ ⇒$$5m2=15$$ ⇒$$m2=3$$

Questions

Let L be a common tangent line to the curves 4x² + 9y² = 36 and (2x)² + (2y)² = 31. Then the square of the slope of the line L is_____________

Unlock the full solution & master the concept.

Get a detailed solution and exclusive access to our masterclass to ensure you never miss a concept

By Expert Faculty of Sri Chaitanya

NEW

Ready to Test Your Skills?

Check Your Performance Today with our Free Mock Tests used by Toppers!

detailed solution

Given : Ellipse is $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$

$l e t \tan g e n t b e y = m x \pm \sqrt{9 m^{2} + 4} I t i s \tan g e n t t o t h e c i r c l e x^{2} + y^{2} = \frac{31}{4} t h e n \Rightarrow \frac{|\sqrt{9 m^{2} + 4}|}{\sqrt{m^{2}} + 1} = \frac{\sqrt{31}}{2} \Rightarrow 4 (9 m^{2} + 4) = 31 (m^{2} + 1) \Rightarrow 36 m^{2} - 31 m^{2} = 31 - 16 \Rightarrow 5 m^{2} = 15 \Rightarrow m^{2} = 3$