If the parabola y = alpha x^{2} - 6x + beta p | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given the parabola equation: $y = \alpha x^{2} - 6x + \beta \dots (i)$ Since the parabola passes through the point $(0, 2)$,we substitute $x = 0$

Vedclass · Accepted Answer

$\alpha = 2, \beta = 2$

Vedclass · Answer

(C) Given the parabola equation: $y = \alpha x^{2} - 6x + \beta \dots (i)$ Since the parabola passes through the point $(0, 2)$,we substitute $x = 0$ and $y = 2$ into equation $(i)$: $2 = \alpha(0)^{2} - 6(0) + \beta \implies \beta = 2$ Now,differentiate equation $(i)$ with respect to $x$: $\frac{dy}{dx} = 2\alpha x - 6$ The tangent at $x = \frac{3}{2}$ is parallel to the $X$-axis,which means the slope $\frac{dy}{dx} = 0$ at $x = \frac{3}{2}$: $\left(\frac{dy}{dx}\right)_{x = 3/2} = 2\alpha \left(\frac{3}{2}\right) - 6 = 0$ $3\alpha - 6 = 0 \implies 3\alpha = 6 \implies \alpha = 2$ Thus,the values are $\alpha = 2$ and $\beta = 2$.

If the parabola $y = \alpha x^{2} - 6x + \beta$ passes through the point $(0, 2)$ and has its tangent at $x = \frac{3}{2}$ parallel to the $X$-axis,then:

Similar Questions

$A$ line passes through the point $(-1, 1)$ and makes an angle $\sin^{-1}(\frac{3}{5})$ in the positive direction of the $x$-axis. If this line meets the curve $x^2 = 4y - 9$ at points $A$ and $B$, then the length $|AB|$ is equal to

The equation of a tangent to the parabola $y^2 = 4ax$ making an angle $\theta$ with the $x$-axis is

Let $P(\alpha, \beta)$ be a point on the parabola $y^2 = 4x$. If $P$ also lies on the chord of the parabola $x^2 = 8y$ whose midpoint is $(1, 5/4)$,then $(\alpha - 28)(\beta - 8)$ is equal to:

The locus of the mid-point of the line segment joining the focus of the parabola $y^{2}=4ax$ to a moving point of the parabola,is another parabola whose directrix is

The length of the chord of the parabola $y^{2}=4ax$ $(a>0)$ which passes through the vertex and makes an acute angle $\alpha$ with the axis of the parabola is

Vedclass Test Series

Exam Paper Generator

Online Exam Module