Question

The value of 'a' for which y = x² + ax + 25 touches the axis of x are ______________.

Answer 1

The given curve is y = x² + ax + 25.

Suppose this curve touches the x-axis at (h, 0).

∴ 0 = h² + ah + 25 .....(1)

y = x² + ax + 25

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=2x+a$

∴ Slope of tangent at (h, 0) = ${\left(\frac{dy}{dx}\right)}_{\left(h,0\right)}=2h+a$

The given curve touches the x-axis at (h, 0). Therefore,

${\left(\frac{dy}{dx}\right)}_{\left(h,0\right)}=0$ (Slope of the x-axis = 0)

$\Rightarrow 2h+a=0$

$\Rightarrow h=-\frac{a}{2}$

Putting $h=-\frac{a}{2}$ in (1), we get

$0={\left(-\frac{a}{2}\right)}^2+a\times \left(-\frac{a}{2}\right)+25$

$\Rightarrow \frac{a^2}{4}-\frac{a^2}{2}+25=0$

$\Rightarrow \frac{a^2}{4}=25$

$\Rightarrow a^2=100$

$\Rightarrow a=\pm 10$

So, the values of 'a' are −10 and 10.

Thus, the values of 'a' for which y = x² + ax + 25 touches the x-axis are −10 and 10.


The value of 'a' for which y = x² + ax + 25 touches the axis of x are __−10 and 10__.