There are two quadratic functions \(f(x),~g(x)\) and a linear function \(h(x).\) \(f(x)\) and \(h(x)\) contact with each other at only one point, \((\alpha,~f(\alpha)).\) \(g(x)\) and \(h(x)\) contact with each other at only one point, \((\beta,~h(\beta)).\)

They satisfy the below conditions:

- Leading coefficients of \(f(x)\) and \(g(x)\) are \(1\) and \(4,\) respectively.
- Two positives \(\alpha\) and \(\beta\) satisfy \(\alpha:\beta=1:2.\)

Let \(t\) be the \(x\)-coordinate of the point of intersection of \(y=f(x)\) and \(y=g(x)\) whose \(x\)-coordinate is in between \(\alpha\) and \(\beta.\)

Find the value of \(\dfrac{210t}{\alpha}.\)

*This problem is a part of <Grade 10 CSAT Mock test> series.*

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