If line $$y=x+2$$ does not intersect any member of family of parabolas $$y2=ax$$,a∈$$R+at$$ two distinct points, then the maximum length of latus rectum of parabola is

Given family of parabola is $$y2=ax$$ Given line is $$y=x+2solving(x+2)2$$−$$ax=0$$⇒ $$x2+x(4$$−$$a)+4=0$$ According to question D≤$$0$$⇒ a≤$$8$$

Parabola

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Question

$If line y = x + 2 does not intersect any member of family of parabolas y^{2} = a x, (a \in R^{+}) at two distinct points,$
$then the maximum length of latus rectum of parabola is$

Moderate

Solution

$Given family of parabola is y^{2} = a x$
$Given line is y = x + 2$
solving
$\begin{array}{l} (x + 2)^{2} - a x = 0 \\ \Rightarrow x^{2} + x (4 - a) + 4 = 0 \\ According to question \\ D \leq 0 \\ \Rightarrow a \leq 8 \end{array}$

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