Question

If the straight line x cosα + y sinα = p is tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). then 

1. \(p^2 = \frac{a^2b^2}{4}\)
2. p2 = a2 cos2 α - b2 sin3. p2 = a2 cos2 α + b2 sin4. None of these

Answer 1

Correct Answer - Option 3 : p2 = a2 cos2 α + b2 sin Concept: If a line y = mx + c     ------ (1) Touches the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)  then c² = a²m² + b²          -------(2) Calculation: Given – line xcos α + ysin α = 0    -------(A) And ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)  from equation (A) y sinα = - xcos α + b \(y = - x\frac{{\cos \alpha }}{{\sin \alpha }} + \frac{b}{{\sin \alpha }}\) On comparing with (1),  \(m = - \frac{{\cos \alpha }}{{\sin \alpha }}\) and \(c = \frac{b}{{\sin \alpha }}\) Then from equation (2) c² = a²m² + b² \(\frac{{{b^2}}}{{{{\sin }^2}\alpha }} = {a^2}\left( {\frac{{{{\cos }^2}\alpha }}{{{{\sin }^2}\alpha }}} \right) + {b^2}\) b² = a²cos² α + b² sin² α