Tangent Point D86Ef1

1. **State the problem:** We need to find the point where the tangent line $4x + 3y = 26$ touches the circle centered at $(-2, 3)$. 2. **Recall the formula for the distance from a point to a line:** The distance $d$ from a point $(x_0, y_0)$ to the line $Ax + By + C = 0$ is given by: $$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$ 3. **Identify the radius of the circle:** Since the line is tangent to the circle, the distance from the center to the line equals the radius $r$ of the circle. 4. **Rewrite the line in standard form:** $$4x + 3y = 26 \implies 4x + 3y - 26 = 0$$ So, $A=4$, $B=3$, and $C=-26$. 5. **Calculate the distance from center $(-2,3)$ to the line:** $$d = \frac{|4(-2) + 3(3) - 26|}{\sqrt{4^2 + 3^2}} = \frac{|-8 + 9 - 26|}{\sqrt{16 + 9}} = \frac{|-25|}{5} = \frac{25}{5} = 5$$ So, the radius $r = 5$. 6. **Find the equation of the circle:** $$ (x + 2)^2 + (y - 3)^2 = 5^2 = 25 $$ 7. **Find the point(s) of intersection between the line and the circle:** From the line equation: $$4x + 3y = 26 \implies 3y = 26 - 4x \implies y = \frac{26 - 4x}{3}$$ Substitute into the circle equation: $$ (x + 2)^2 + \left(\frac{26 - 4x}{3} - 3\right)^2 = 25 $$ Simplify the term inside the second square: $$ \frac{26 - 4x}{3} - 3 = \frac{26 - 4x - 9}{3} = \frac{17 - 4x}{3} $$ So, $$ (x + 2)^2 + \left(\frac{17 - 4x}{3}\right)^2 = 25 $$ Multiply both sides by $9$ to clear denominators: $$ 9(x + 2)^2 + (17 - 4x)^2 = 225 $$ Expand: $$ 9(x^2 + 4x + 4) + (17 - 4x)^2 = 225 $$ $$ 9x^2 + 36x + 36 + (17 - 4x)^2 = 225 $$ Expand $(17 - 4x)^2$: $$ (17)^2 - 2 \times 17 \times 4x + (4x)^2 = 289 - 136x + 16x^2 $$ So, $$ 9x^2 + 36x + 36 + 289 - 136x + 16x^2 = 225 $$ Combine like terms: $$ (9x^2 + 16x^2) + (36x - 136x) + (36 + 289) = 225 $$ $$ 25x^2 - 100x + 325 = 225 $$ Subtract 225 from both sides: $$ 25x^2 - 100x + 100 = 0 $$ Divide entire equation by 25: $$ \cancel{25}x^2 - \cancel{100}x + \cancel{100} = 0 \implies x^2 - 4x + 4 = 0 $$ This factors as: $$ (x - 2)^2 = 0 $$ So, $$ x = 2 $$ 8. **Find corresponding $y$ value:** $$ y = \frac{26 - 4(2)}{3} = \frac{26 - 8}{3} = \frac{18}{3} = 6 $$ 9. **Final answer:** The tangent point is at $$ \boxed{(2, 6)} $$