Name: Applications of Section Formula
Uploaded: 2019-07-19
Duration: 14 min 56 s
Description: Thorough with the section formula in three-dimensional geometry? This video will show you how the section formula can be applied in various situations in geo...

Question

A line has equation $2 x + y = 20$ and a curve has equation $y = a + \frac{18}{x - 3}$ , where $a$ is a constant. Find the set of values of $a$ for which the line does not intersect the curve.

Accepted Answer

Given line is $2 x + y = 20$ ...(i)

Given curve has equation $y = a + \frac{18}{x - 3}$ ...(ii)

Substituting (ii) in (i),

$\Rightarrow 2 x + a + \frac{18}{x - 3} = 20$

$\Rightarrow 2 x (x - 3) + a (x - 3) + 18 = 20 (x - 3)$

$\Rightarrow 2 x^{2} - 6 x + a x - 3 a + 18 = 20 x - 60$

$\Rightarrow 2 x^{2} + a x - 26 x + 78 - 3 a = 0$

$\Rightarrow 2 x^{2} + (a - 26) x + (78 - 3 a) = 0$

For no real roots, $b^{2} - 4 a c < 0$ , where $a = 2, b = a - 26, c = 78 - 3 a$

$\Rightarrow (a - 26)^{2} - 4 (2) (78 - 3 a) < 0$

$\Rightarrow a^{2} - 52 a + 676 - 624 + 24 a < 0$

$\Rightarrow a^{2} - 28 a + 52 < 0$

A graph of $y = a^{2} - 28 a + 52$ is a $\cup$ shaped parabola.

$\Rightarrow a^{2} - 28 a + 52 = 0$

$\Rightarrow (a - 2) (a - 26) = 0$

$\Rightarrow a = 2$ or $a = 26$

For $a^{2} - 28 a + 52 < 0$ we need to find the range of values of $x$ for which the curve is negative (below the $a -$ axis)

The solution is $2 < a < 26$

Therefore, the set of values of $a$ for which the line does not intersect the curve is $2 < a < 26$ .

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