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Solve Lesson 7 Page 67 Math Study Topic 10 – Kite>

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Topic

Given a line \(\Delta \) and a point O such that the distance from O to \(\Delta \) is OH = 1 (Figure 39). For each moving point M in the plane, let K be the perpendicular projection of M onto \(\Delta \). Prove the set of points M in the plane such that \(M{K^2} – M{O^2} = 1\) is a parabola.

Detailed explanation

Choose the coordinate system so that the point O coincides with the origin and the axis Ox coincides with the line OH.

Assuming M has coordinates (x; y), then K has coordinates (-1; y).

Then:

\(\begin{array}{l}M{K^2} – M{O^2} = 1\\ \Rightarrow {\left( {x + 1} \right)^2} + {\left( { y – y} \right)^2} – {\left( {0 – x} \right)^2} – {\left( {0 – y} \right)^2} = 1\\ \Rightarrow {x ^2} + 2x + 1 – {x^2} – {y^2} = 1 \Rightarrow {y^2} = 2x\end{array}\)

So the set of points M is a parabola with the equation \({y^2} = 2x\)

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