If 16a² + 25b²- c² = 40ab, then the family of lines ax + by + c = 0 is concurrent at the points)
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Please find below the solution to the asked query:
$$\begin{gathered} \Rightarrow 16{\mathrm{a}}^2+25{\mathrm{b}}^2-{\mathrm{c}}^2=40\mathrm{ab} \\ \Rightarrow 16{\mathrm{a}}^2+25{\mathrm{b}}^2-40\mathrm{ab}-{\mathrm{c}}^2=0 \\ \Rightarrow {\left(4\mathrm{a}-5\mathrm{b}\right)}^2-{\mathrm{c}}^2=0 \\ \mathrm{Use} {\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}-\mathrm{b}\right) \left(\mathrm{a}+\mathrm{b}\right) \\ \Rightarrow \left(4\mathrm{a}-5\mathrm{b}-\mathrm{c}\right) \left(4\mathrm{a}-5\mathrm{b}+\mathrm{c}\right)=0 \\ \mathrm{So} \\ \left(4\mathrm{a}-5\mathrm{b}-\mathrm{c}\right)=0 \mathrm{or} \left(4\mathrm{a}-5\mathrm{b}+\mathrm{c}\right)=0 \\ \mathrm{Family} \mathrm{of} \mathrm{lines} \mathrm{is} \mathrm{ax}+\mathrm{by}+\mathrm{c}=0 \\ \mathrm{So} \\ \mathrm{x}=4 \mathrm{and} \mathrm{y}=-5 \mathrm{satisfythe} \mathrm{family} \mathrm{of} \mathrm{lines} \mathrm{thus} \mathrm{the} \mathrm{given} \mathrm{lines} \mathrm{are} \mathrm{concurrent} \mathrm{at} \mathrm{point} \left(4,-5\right) \left(\mathrm{Answer}\right) \end{gathered}$$

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