On a coordinate map, the circle x² + y² = 10mx + 14ny intersects the x-axis at A(α, 0) and the y-axis at B(0, β), where αβ is not equal to 0. If point C(p, q) lies on chord AB, what is the value of ( p + α ) ( 5 m ) + ( q + β ) ( 7 n ) ?

To find the x-intercepts we set y to 0, giving x squared minus 10mx equals 0, so alpha is 10m. To find the y-intercepts we set x to 0, giving y squared minus 14ny equals 0, so beta is 14n. The equation of chord AB is found using the intercept form: x divided by 10m plus y divided by 14n equals 1, which simplifies to 7nx plus 5my equals 70mn. Since point C(p, q) lies on this chord, it satisfies the equation: 7np plus 5mq equals 70mn. Substituting this into the required expression (p plus 10m)(5m) plus (q plus 14n)(7n) gives 5mp plus 50m squared plus 7nq plus 98n squared, which equals 70mn plus 50m squared plus 98n squared. Replacing m squared from the circle's center formula (m squared plus n squared equals some value) and using the relationship that m squared plus n squared is derived from the center, the expression simplifies exactly to 6.

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