# #21 of Sept 2016 Grade 10 CSAT(Korean SAT) mock test

Algebra Level 3

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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