- Geometric Application of determinants: (Curves and surfaces through given points)The idea is to get an equation from the vanishing of the determinant of a homogeneous linearsystem as the condition for a nontrivial solution. We explain the trick for obtainingsuch a system for the case of a line L through two given points P1 : (x1 yı)and P2 : (x2.yz). The unknown line is ax +by+c=0, say. We write it as ax +by+c.1=0. To get a nontrivial solution for a,b,c, the determinant of the"coefficients” x,y, 1 must be zero. The system isaxtby +c. 1 =0 (Line L)axı +by. +0.1=0 (P1 on L)... (1)ax2 +by2 +0.1=0 (P2 on L)(a) Line through two given points: Derive from D=0 in (1) the familiarformula(2-1))(1-x)=O'-1)Ya-D(b) Plane : Find the analog of (1) for a plane through three given points. Apply it when thepoints are(1,1,1), (3,2,6), (5,0,5)(c) Circle : Find a similar formula for a circle in the plane through three given points. Findand sketch the circle through (2,6), (6,3), (7,1).(d) Sphere: Find the analog of the formula in (C) for a sphere through four given points. Findthe sphere through (0,0,5), (4,0,1), (0,4,1), (0,0,-3) by the formula or byinspection​