# Equations of Conic Sections Systems of Non-linear Equations Solve eeach problem systematically 1. Find all values of m so that the graph 2mx^{2} - 16mx + my^{2} + 7y^{2} = 2m^{2} - 18m is a circle.

Equations of Conic Sections Systems of Non-linear Equations Solve eeach problem systematically 1. Find all values of m so that the graph $$2mx^{2}\ -\ 16mx\ +\ my^{2}\ +\ 7y^{2} = 2m^{2}\ -\ 18m$$ is a circle.

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Gennenzip
Given equation is, $$2mx^{2}\ −\ 16mx\ +\ my^{2}\ +\ 7y^{2} = 2m^{2}\ −\ 18m$$ Which can be written as, $$x^{2}\ -\ 8x\ +\ \frac{m\ +\ 7}{2n}y^{2}=m\ -\ 9$$
$$\Rightarrow\ x^{2}\ -\ 8x\ +\ 16\ +\ \frac{m\ +\ 7}{2m}(y\ -\ 0)^{2}=m\ -\ 9\ +\ 16$$
$$\Rightarrow(x\ -\ 4)^{2}\ +\ \frac{m\ +\ 7}{2m}(y\ -\ 0)^{2}=m\ +\ 7$$ Equation of circle is given as, $$(x\ −\ a)^{2}\ +\ (y\ −\ b)^{2} = r^{2}$$ If the given equation is an equation of circle, then $$\frac{m\ +\ 7}{2m}\ \text{should be equal to} 1\ and\ m\ +\ 7\ >\ 0$$ So, $$\frac{m\ +\ 7}{2m}=1$$
$$m\ +\ 7 = 2m$$
$$m =7$$ And, $$m\ +\ 7 = 14\ >\ 0,$$ Therefore the graph $$2\ mx^{2}\ -\ 16\ mx\ +\ my^{2}\ +\ 7y^{2} = 2m^{2}\ -\ 18m\ \text{will be a circle if}\ m = 7$$