x2+y2=a2 is therefore the equation of a circle with the radius a having its center at the origin. Again, let the center of the circle be at the given point (b, c), and let P(x, y) be any point of the circle. √(x−b)2+(y−c)2 = a or. (x-b)2+(y—c)2=a2 is the equation of a circle with the radius a and the center (b, c). This equation, when simplified, is x2+ y2 — 2bx-2cy—a2+b2+c2=0; it has the peculiarities that it is of the second degree, but has no xy term, and that the coefficients of x2 and y2 are equal. Any equation having these peculiarities is the equation of a circle, and its locus is easily identified. For instance, 2x2+2y2-6x+10y+9=0, by adding to both sides enough to complete the two squares, may be written in the typical form as (x − 3)2+(y+5)2=4; hence the equation represents a circle with the radius 2 and the center (,-). Examples. 1. Find the locus of points equally distant from the axes of coordinates. 2. Find the locus of points equally distant from 2x-3y+5=0 and 3x-2y+6=0. Ans. (2x-3y+5)=±(3x-2y+6); x+y+1=0 or 5x-5y+11=0. 3. Find the locus of points five units from (-2, 1). 4. Find the locus of points equidistant from (−3,1) and (3, 3) by the process Art. 109, and check by finding the equation of the mid-perpendicular to the line joining the given points. 5. The base of a triangle is the part of the x-axis from (−3,0) to (2, 0), and the eter of its vertical angle passes through the origin. Find the equation of the locus of the vertex, and identify it. (Use the theorem about the segments into which the base of a triang' is divided by the bisector of the vertical angle.) 6. Locate the cent and give the radius of x2 + y2 −12x+1y+12=0; x2+ y2+8x-2y-8=0. 7. Find the equation of a circle having the center (1, -1) and passing through 12,2). 111. Simultaneous Equations.-We can now represent graphically the simultaneous solution of any linear equation and a quadratic equation of the type ax2+ay2+ dx+ey+f=0, or of two such quadratics. These solutions correspond, as we have seen, to the coordinates of the intersection of the loci of the two equations. The real roots can be approximated by measuring the coordinates of the intersections, and the figure will show the nature of the roots. Two pairs of roots are to be expected; if the loci do not intersect, these roots are imaginary, and if the loci are tangent, so that two intersections have merged, the two pairs of roots will be alike. V The quadratic terms can be eliminated from any pair of equations of the type ax2+ay2+ dx+ey+f=0, and if we solve the resulting linear equation with either of the given quadratic equations, we shall get the simultaneous solutions of the two quadratics. must also satisfy (2) and not the (1) and (2) represent circles and (3) represents a straight line. Any point satisfying both (1) and (2) (3); but any point satisfying either (1) or other cannot satisfy (3). Hence the locus of (3) meets the loci of (1) and (2) at their common points, but not anywhere else, and so is the common chord of the two circles. (See Fig. 19.) 112. The principle brought out here is general, and may be stated: k and 7 being constant, kf(x, y) −14(x, y) =0 represents a curve passing through all the common points of f(x, y) =0 and (x, y) = 0, and not meeting either at any other point; for a point common to both makes f(x, y) and (x, y) both zero, and a point that is on one and not on the other, makes either f(x, y) or p(x, y) zero, but not both. A related principle is : f(x, y)×$(x,y) =0 represents the two curves, f(x, y) =0 and (x, y) = 0, together; for any point of either curve makes f(x, y) or (x, y) equal to zero and so makes the product zero. Thus (2x-3y+1) (3x+y-2)=6x2-7xy-3y2-x+7y-2=0 represents two straight lines; xy=0 represents the coordinate axes; x2-y2=0 represents the bisectors of the angles between the axes. Examples. Illustrate by graphs the solution of the following simultaneous equations (check by solving and plotting the solutions): 1. x2+y2+6x-6y-47-0 with each of the following in turn (draw one circle cut by the various lines): (a) y-x+5=0, (b) y-x+1=0, (c) y+x=0, (f) 8x+y=44. 2. x2+y2-10x-8y-24-0 with y+x=0. x2+y2+6x-6y-47-0 with x2+y2-10x+2y+1=0. x2+y2+6x-6y-47-0 with x2+y2−14x+4y+13=0. 5. x2+ y2-2x-4y=0 with x2+y2 −14x+2y-30=0. 113. Derivatives of y2 and xy.—When the derivatives of more than one function are used at the same time, we are obliged to add to our notation. As we saw in the discussion of Art. 84, the derivative with respect to x of a given function of x is the slope of the tangent to the graph of the function, and is determined by finding the limit of the slope of a secant line as it swings into coincidence with the tangent. If the function treated is represented by y, the increase given to x (PM, Fig. 20) is represented by dx, a compound symbol meaning " differential of x"; the length MT, the height of the tangent line above the point M, is represented by dy, meaning “ differential of y." Thus the derivative of y with respect to x, defined as the MT value of is represented by the symbol dy, which is ordi PM dx narily read either "dy over dx" or "the derivative of y with respect to x." In other words, when f(x) is represented by y, |