If y=-2 is a tangent ⇒px^2+qx+r+2=0 will have only one real root ⇒q^2-4 p(r+2)=0 =q^2-4 pr-8 p

step 1 Y equal minus 2 is a tangent and given that px square plus QX plus R plus plus R plus 2 equal 0

If y=-2 is a tangent ⇒px^2+qx+r+2=0 will have only one...

If $\mathrm{y}=-2$ is a tangent $\Rightarrow \mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}+2=0 \quad$ will have only one real root $\Rightarrow \mathrm{q}^{2}-4 \mathrm{p}(\mathrm{r}+2)=0$ $=\mathrm{q}^{2}-4 \mathrm{pr}-8 \mathrm{p}=0$ $\Rightarrow \mathrm{p}<0$ as $\left(\mathrm{q}^{2}-4 \mathrm{pr}<0\right)$ $\Rightarrow \mathrm{r}<0$

If $\mathrm{y}=-2$ is a tangent
$\Rightarrow \mathrm{px}^{2}+\mathrm{qx}+\mathrm{r}+2=0 \quad$ will have only one real root
$\Rightarrow \mathrm{q}^{2}-4 \mathrm{p}(\mathrm{r}+2)=0$
$=\mathrm{q}^{2}-4 \mathrm{pr}-8 \mathrm{p}=0$
$\Rightarrow \mathrm{p}<0$ as $\left(\mathrm{q}^{2}-4 \mathrm{pr}<0\right)$
$\Rightarrow \mathrm{r}<0$

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Key Concepts

Tangency Condition

A tangency condition between a line and a curve occurs when the line touches the curve at exactly one point of intersection. In the context of a quadratic function, this means that when the equation of the line is substituted into the quadratic, the resulting equation has a unique solution.

Quadratic Discriminant

The discriminant of a quadratic equation, given by the expression b² - 4ac, determines the nature and number of its roots. A discriminant equal to zero indicates that the quadratic has one repeated (or double) real root, which is essential for establishing tangency between a quadratic curve and a line.

Coefficient Constraints and Parabola Orientation

The coefficients of a quadratic function influence the shape and orientation of its graph. In problems involving tangency, certain inequalities or conditions on these coefficients may be derived to ensure the curve behaves in a specific manner, such as opening upward or downward, which is crucial for the line to just touch the curve.

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