Solve the following locus of points problems. (a) The set of points equidistant from two intersecting lines. Hint: Why can we let the lines be $y = 0$ and $y = mx$ or $y = 0$ and $x = 0$? The distance from a point to a line is given by the distance between the point and the intersection of the line and the perpendicular to it through the point. (b) The set of points whose distance from one line is half the distance from another line. (c) The set of points whose distance from one line is $k$ times the distance from another line. (d) The set of points $P$ so that the distance of $P$ to $y = 0$ times the distance from P to $x = b$ equals the square of the distance of $P$ from $y = x$. (This problem is a special case of the three line locus problem worked on by the Greeks and completely solved by Descartes.)
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This is because the distance from a point to a line is given by the distance between the point and the intersection of the line and the perpendicular to it through the point. Show more…
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Show that the points $P=(a, b)$ and $Q=(b, a)$ are symmetric with respect to the line $y=x$ as follows. (a) Find the slope of the line through $P$ and $Q$ (b) Use slopes to show that the line through $P$ and $Q$ is perpendicular to $y=x$ (c) Let $R$ be the point where the line $y=x$ intersects line segment $P Q .$ since $R$ is on $y=x,$ it has coordinates (c, c) for some number $c,$ as shown in the figure. Use the distance formula to show that segment $P R$ has the same length as segment $R Q .$ Conclude that the line $y=x$ is the perpendicular bisector of segment $P Q$ Therefore, $P$ and $Q$ are symmetric with respect to the line $y=x$ GRAPH CANT COPY
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Consider the problem of fitting a conic through $m$ given points $P_{1}\left(x_{1}, y_{1}\right), \dots$ $\boldsymbol{P}_{m}\left(\boldsymbol{x}_{\boldsymbol{m}}, \boldsymbol{y}_{\boldsymbol{m}}\right)$ in the plane; see Exercises 53 through 62 in Section 1.2. Recall that a conic is a curve in $\mathbb{R}^{2}$ that can be described by an equation of the form $f(x, y)=$ $c_{1}+c_{2} x+c_{3} y+c_{4} x^{2}+c_{5} x y+c_{6} y^{2}=0,$ where at least one of the coefficients $c_{i}$ is nonzero. Explain why fitting a conic through the points $P_{1}\left(x_{1}, y_{1}\right), \ldots, P_{m}\left(x_{m}, y_{m}\right)$ amounts to finding the kernel of an $m \times 6$ matrix $A$. Give the entries of the $i$ th row of $A$ Note that a one-dimensional subspace of the kernel of $A$ defines a unique conic, since the equations $f(x, y)=0$ and $k f(x, y)=0$ describe the same conic.
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