Question

1. -/2 points details my notes ask your teacher practice another draw a diagram to show that there are two tangent lines to the parabola of y = x² that pass through the point (0, -25). find the coordinates of the points where these tangent lines intersect the parabola. smaller x - value (x, y) = ( ) larger x - value (x, y) = ( )

Explanation:

Step1: Find the derivative of the parabola

The derivative of $y = x^{2}$ is $y'=2x$. Let the point of tangency on the parabola be $(a,a^{2})$. The slope of the tangent - line at $x = a$ is $m = 2a$.

Step2: Use the point - slope form of a line

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$. The line passes through $(a,a^{2})$ and $(0,-25)$, so the slope $m=\frac{a^{2}+25}{a}$. Since $m = 2a$, we have the equation $\frac{a^{2}+25}{a}=2a$.

Step3: Solve the equation for $a$

Cross - multiply the equation $\frac{a^{2}+25}{a}=2a$ to get $a^{2}+25 = 2a^{2}$. Rearranging gives $a^{2}=25$, so $a=\pm5$.

Step4: Find the coordinates of the points of tangency

When $a=-5$, $y=a^{2}=25$, and the point is $(-5,25)$. When $a = 5$, $y=a^{2}=25$, and the point is $(5,25)$.

Answer:

smaller $x$-value $(x,y)=(-5,25)$
larger $x$-value $(x,y)=(5,25)$