If \(\mathrm{x}+\mathrm{y}+1=0\) touches the parabola \(\mathrm{y}^{2}=\mathrm{ax}\) then \(\mathrm{a}=\ldots \ldots \ldots\) (a) 8 (b) 6 (c) 4 (d) 2

When dealing with quadratic equations, the discriminant is a powerful tool that tells us about the nature of the roots without actually solving the equation. Given a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by the expression \( D = b^2 - 4ac \).

The value of the discriminant reveals whether the quadratic equation has two distinct real roots (if \( D > 0 \) ), one repeated real root (if \( D = 0 \) ), or no real roots (if \( D < 0 \) ). In the context of a line tangent to a parabola, a discriminant of zero means that the line just touches the parabola at one point and does not cross it, indicating a perfect tangent. Students should remember that when they find a discriminant equal to zero, they can be confident that there's precisely one point of tangency between the line and the quadratic curve.

The slope of a line is a measure of its steepness, often denoted as \( m \). In the coordinate plane, it can be calculated as the change in \( y \) divided by the change in \( x \) between two distinct points on the line. The slope is fundamental when studying lines because it allows us to determine the angle and direction of a line, as well as to find equations of lines. A positive slope means the line ascends from left to right, while a negative slope means it descends. A horizontal line has a slope of zero, and a vertical line's slope is undefined because the change in \( x \) is zero, which would make the calculation involve division by zero.

When a line is said to be tangent to a curve, such as a parabola, the slope of the line at the point of tangency is equal to the slope of the tangent to the curve at that same point. This is key to solving problems like finding a specific value that will make a line tangent to a parabolic shape.

Differentiation of parametric equations involves finding the rate of change in terms of a parameter, which typically represents time or some other variable that both expressions are dependent upon. Unlike traditional functions where \( y \) is expressed explicitly in terms of \( x \) (i.e., \( y = f(x) \) ), parametric equations define both \( x \) and \( y \) in terms of a third variable, often denoted as \( t \).

To differentiate parametric equations \( x = f(t) \) and \( y = g(t) \), you compute \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) separately. Then, to find the slope of the tangent line at a particular point, you use the ratio \( \frac{dy/dt}{dx/dt} \), which gives you \( \frac{dy}{dx} \). When working with curves defined parametrically, such as parabolas or ellipses, understanding how to differentiate with respect to a parameter is crucial to determining the nature of the curve at any given point, which is especially important for identifying slopes of tangents, areas, or lengths related to the shape.