Question

verify that the given point is on the curve and find the lines that are a. tangent and b. normal to the curve at the given point.
2x² - √5xy + y² = 6, (√5,4)
the point is on the curve because when √5 is substituted for x and 4 is substituted for y, the resulting statement is 6 = 6, which is a true statement.
(simplify your answers.)
a. the equation of the tangent line to the curve at the point (√5,4) is y = 4
b. the equation of the normal line to the curve at the point (√5,4) is

Explanation:

Step1: Recall relationship between tangent and normal

The normal line is perpendicular to the tangent line. If the slope of the tangent line $m_t$ is known, the slope of the normal line $m_n$ satisfies $m_t\times m_n=- 1$.

Step2: Determine slope of tangent line

The equation of the tangent line at $(\sqrt{5},4)$ is $y = 4$, which is a horizontal line. The slope of a horizontal line $m_t=0$.

Step3: Calculate slope of normal line

Since $m_t\times m_n=-1$ and $m_t = 0$, the slope of the normal line $m_n$ is undefined. A line with an undefined slope is a vertical line.

Step4: Find equation of normal line

The equation of a vertical line passing through the point $(x_0,y_0)=(\sqrt{5},4)$ is of the form $x = x_0$. So the equation of the normal line is $x=\sqrt{5}$.

Answer:

$x=\sqrt{5}$