Ellipse
Ellipse
nta_abhyas_2025
Grade 11
Question:
Let the line $x = my$ and the ellipse $2x^2 + y^2 = 1$ intersect at a point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $\left(-\frac{\sqrt{3}}{2}, 0\right)$ and $(0, \beta)$, then $\beta$ is equal to
\frac{2\sqrt{3}}{3}
\frac{4\sqrt{3}}{3}
\frac{2}{3}
\frac{4}{3}
Step-by-Step Solution
Key Concept: Two equations represent the same line when their coefficients are proportional
From $4lx - ky = 5$ being the same as $2x + y = 3$, we equate coefficients: $ rac{4l}{2} = rac{-k}{1} = rac{5}{3}$. This gives $4l = rac{10}{3}$ so $l = rac{5}{6}$, and $-k = rac{5}{3}$ so $k = - rac{5}{3}$. The point of intersection of the tangents is $( rac{1}{2}, rac{1}{3})$.
Correct Answer: 1