Question

1. -/2 points find equations of both the tangent lines to the ellipse x² + 8y² = 72 that pass through the point (24, 3). smaller slope y = larger slope y =

Explanation:

Step1: Differentiate the ellipse equation implicitly

Differentiate $x^{2}+8y^{2}=72$ with respect to $x$.
Using the power - rule and chain - rule, we have $2x + 16y\frac{dy}{dx}=0$. Then $\frac{dy}{dx}=-\frac{x}{8y}$.
Let the point of tangency on the ellipse be $(x_0,y_0)$. The slope of the tangent line at this point is $m = -\frac{x_0}{8y_0}$.
The equation of the tangent line using the point - slope form $y - y_1=m(x - x_1)$ (where $(x_1,y_1)=(x_0,y_0)$) is $y - y_0=-\frac{x_0}{8y_0}(x - x_0)$.
Since the tangent line passes through the point $(24,3)$, we substitute $x = 24$ and $y = 3$ into the tangent - line equation: $3 - y_0=-\frac{x_0}{8y_0}(24 - x_0)$.
Also, since $(x_0,y_0)$ lies on the ellipse $x_0^{2}+8y_0^{2}=72$, we have $x_0^{2}=72 - 8y_0^{2}$.

Step2: Simplify the equation

Starting from $3 - y_0=-\frac{x_0}{8y_0}(24 - x_0)$, we multiply both sides by $8y_0$ to get $24y_0-8y_0^{2}=-24x_0 + x_0^{2}$.
Substitute $x_0^{2}=72 - 8y_0^{2}$ into the above equation:
$24y_0-8y_0^{2}=-24x_0+72 - 8y_0^{2}$.
This simplifies to $24y_0+24x_0=72$, or $x_0 + y_0 = 3$, so $x_0=3 - y_0$.

Step3: Solve for $y_0$

Substitute $x_0=3 - y_0$ into the ellipse equation $x_0^{2}+8y_0^{2}=72$:
$(3 - y_0)^{2}+8y_0^{2}=72$.
Expand $(3 - y_0)^{2}$: $9-6y_0+y_0^{2}+8y_0^{2}=72$.
Combine like terms: $9y_0^{2}-6y_0 - 63 = 0$.
Divide through by $3$: $3y_0^{2}-2y_0 - 21 = 0$.
Factor the quadratic equation: $(3y_0 + 7)(y_0 - 3)=0$.
So $y_0=3$ or $y_0=-\frac{7}{3}$.

Step4: Solve for $x_0$

When $y_0 = 3$, from $x_0=3 - y_0$, we get $x_0=0$.
When $y_0=-\frac{7}{3}$, from $x_0=3 - y_0$, we get $x_0=3+\frac{7}{3}=\frac{9 + 7}{3}=\frac{16}{3}$.

Step5: Find the slopes

When $(x_0,y_0)=(0,3)$, the slope $m_1=-\frac{x_0}{8y_0}=0$.
When $(x_0,y_0)=(\frac{16}{3},-\frac{7}{3})$, the slope $m_2=-\frac{\frac{16}{3}}{8\times(-\frac{7}{3})}=\frac{2}{7}$.

Step6: Find the equations of the tangent lines

For the smaller slope $m = 0$, using the point - slope form with the point $(24,3)$, the equation of the tangent line is $y-3=0(x - 24)$, so $y = 3$.
For the larger slope $m=\frac{2}{7}$, using the point - slope form $y - 3=\frac{2}{7}(x - 24)$.
Expand to get $y-3=\frac{2}{7}x-\frac{48}{7}$, then $y=\frac{2}{7}x-\frac{48}{7}+3=\frac{2}{7}x-\frac{48 - 21}{7}=\frac{2}{7}x-\frac{27}{7}$.

Answer:

smaller slope: $y = 3$
larger slope: $y=\frac{2}{7}x-\frac{27}{7}$