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  4. An ellipse with focii (1,4) and (α, β) touches x-axis at (5,0). Then value of (α-β) is

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Q. An ellipse with focii $(1,4)$ and $(\alpha, \beta)$ touches $x$-axis at $(5,0)$. Then value of $(\alpha-\beta)$ is

Conic Sections

Solution:

$(1,4),(5,0)$ and reflection of $(\alpha, \beta)$ on $x$-axis are collinear
$\begin{vmatrix}1 & 4 & 1 \\ 5 & 0 & 1 \\ \alpha & -\beta & 1\end{vmatrix}=0 \Rightarrow \alpha-\beta=5$