ParabolaHardBloom L3
Question
The set of points $(x,y)$ whose distance from the line $y = 2x + 2$ is the same as the distance from $(2,0)$ is a parabola. This parabola is congruent to the parabola in standard form $y = Kx^{2}$ for some $K$ which is equal to :
Options
A.$\frac{\sqrt{5}}{12}$
B.$\frac{\sqrt{5}}{4}$
C.$\frac{4}{\sqrt{5}}$
D.$\frac{12}{\sqrt{5}}$
Solution
Sol. [Key Concept: Equal distance from a fixed point and a line is a parabola; its latus rectum (from that point–line distance) fixes $K$ via $4a = \frac{1}{|K|}$.]
$$x^{2} = \frac{1}{k}y $$
Length of $LR = 2(SM)$
$$\begin{matrix} & \ = 2\frac{|2(2) + 2 - 0|}{\sqrt{2^{2} + 1}} = \frac{12}{\sqrt{5}} \\ \Rightarrow \frac{1}{k} & \ = \frac{12}{\sqrt{5}} \Rightarrow k = \frac{\sqrt{5}}{12} \end{matrix}$$
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