By Chambert-Loir A.

This graduate path has faces: algebra and geometry. certainly, we examine concurrently loci of issues outlined by means of polynomial equations and algebras of finite style over a box. we will convey on examples (Hilbert's Nullstellensatz, size conception, regularity) how those are faces of a unmarried head and the way either geometrie and algebraic features enlight the only the opposite.

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**Example text**

24 and x^-f' = y~ = 4:X and x — y 1 = 0. = 16 and 6y = x^. /- If the equations of the sides of a triangle are 3x +y—7 =0, and x each of the medians. / find the length of : Ch. Ill, § 4. LOCI 21] Which 33 of the points (3, -1), (7, 2), (0, -2), are on the locus of the equation 4 x 5. of X6. / and (8, 3) 14. Find the length of the chord of intersection of the 2/' = 13 and y'~ = 3x-\-3. loci + For what values of b are the two intersections of the y = 2x -\-b and y- = 4x real and distinct ? imaginary ?

34 P, the ordinates the similarity of the two triangles P^KP^ through and (x^^ y^^, OX. and \ Cii. IV, § THE STRAIGHT LINE 23] Substituting these values, we have y-jji _ yi This 35 y\ [5] then the algebraic relation between the coordi- is nates X and ^'i, yi, ^2' y of any point on the line and the constants ^^^ Vv ^^^^ is therefore the equation of the It is called the two-point line. form of the equation of Y the straight line. \^ Let the student show that this equation cannot be satisfied by the coordinates of any point not on the line.

X of in the equation Hence the and finding the corresponding values the intercepts on the ]r-axis, by substituting x ; and finding the corresponding values of 20. Intersection of tersect, tlie in- can be found by substituting tercepts on the X-axis two curves. = y. — When tAvo curves in- coordinates of the point of intersection must satisfy both equations. In- order to find the coordinates of such a point of intersection, it is only necessary to find the values of x and y which will satisfy both equations, or in other words, to solve the equations simultaneously.