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Problem 722

Write the equation of the circle with center \(\mathrm{C}\) at the origin and with radius 7 .

Expert verified

The equation of the circle with center at the origin and radius 7 is \(x^2 + y^2 = 49\).

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Chapter 42

Find the points of intersection (if any) of the circles \(\mathrm{C}_{1}\) and \(C_{2}\) where \(C_{1}: x^{2}+y^{2}-4 x-2 y+1=0\) $$ C_{2}: x^{2}+y^{2}-6 x+4 y+4=0 $$

Chapter 42

Euclid defined a circle as the locus of points equidistant from a given point. Apollonius, on the other hand had an alternate definition: Given two points, \(\mathrm{A}\) and \(\mathrm{B}\), and a constant \(\mathrm{k} \neq 1\), the set of all points P such that \(\mathrm{PA}=\mathrm{k} \cdot \mathrm{PB}\) is a circle. Consider points \(\mathrm{A}(0,0)\) and \(\mathrm{B}(b, 0)\), and the constant \(\mathrm{k}\). Show that the Apollonian definition does indeed lead to the equation of the circle. Also, find the coordinates of the center and the radius of the circle.

Chapter 42

For the equation \(\mathrm{x}^{2} / 25+\mathrm{y}^{2} / 9=1\), find the y coordinates when \(\mathrm{x}=2 ;\) b) \(\mathrm{x}=3 ;\) c) \(\mathrm{x}=4 ;\) d) \(\mathrm{x}=5 ;\) e) \(\mathrm{x}=6\)

Chapter 42

Find the equation of the ellipse which has vertices \(\mathrm{V}_{1}(-2,6)\), \(\mathrm{V}^{2}(-2,-4)\), and foci $\mathrm{F}_{1}(-2,4), \mathrm{F}_{2}(-2,-2)$. (See figure,)

Chapter 42

Given that two circles $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{D}_{1} \mathrm{x}+\mathrm{E}_{1} \mathrm{y}+\mathrm{F}_{1}=0$ and $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{D}_{2} \mathrm{x}+\mathrm{E}_{2} \mathrm{y}+\mathrm{F}_{2}=0$ intersect at two points, show that the equation for the line determined by the points of intersection is $\left(\mathrm{D}_{1}-\mathrm{D}_{2}\right) \mathrm{x}+\left(\mathrm{E}_{1}-\mathrm{E}_{2}\right)+\left(\mathrm{F}_{1}-\mathrm{F}_{2}\right)=0$.

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