Advanced Questions $\newline$ $9$ Find, without differentiation, the equation of the straight lines which pass through the point $(4,0)$ and are tangential to the circle $(x+1)^{2}+y^{2}=4$ .

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Write a quadratic function from its x-intercepts and another point

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Q. Advanced Questions

\newline

9

Find, without differentiation, the equation of the straight lines which pass through the point

(4,0)

and are tangential to the circle

(x+1)^{2}+y^{2}=4

Circle Equation and Radius: The equation of the circle is $(x+1)^{2}+y^{2}=4$ . The radius $r$ of the circle is the square root of $4$ , which is $2$ .
Center and Distance Calculation: The center of the circle is $(-1,0)$ . The distance $d$ from the center of the circle to the point $(4,0)$ is the absolute value of $-1$ minus $4$ , which is $5$ .
Perpendicular Tangent Line: For a line to be tangent to the circle and pass through $(4,0)$ , it must be perpendicular to the radius at the point of tangency. The slope of the radius is the change in $y$ over the change in $x$ between the center of the circle and the point of tangency.
Point-Slope Form of Tangent Line: Since we don't know the exact point of tangency, we can use the fact that the product of the slopes of two perpendicular lines is $-1$ . Let $m$ be the slope of the tangent line. Then, the slope of the radius (which is a line from the center to the point of tangency) is $-\frac{1}{m}$ .
Distance from Center to Tangent Line: The tangent line must pass through $(4,0)$ . Using the point-slope form of a line, $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is $(4,0)$ , we get $y = m(x - 4)$ .
Distance Equation Simplification: The distance from the center of the circle to the tangent line must be equal to the radius of the circle. Using the distance formula for a point to a line, $\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ , where $A = -m$ , $B = 1$ , $C = 4m$ , and $(x_1, y_1)$ is the center $(-1,0)$ , we get $\frac{|(-m)(-1) + (1)(0) + 4m|}{\sqrt{(-m)^2 + 1^2}} = 2$ .
Solving for Slope: Simplifying the distance equation, we get $|m + 4m| / \sqrt{m^2 + 1} = 2$ . This simplifies to $|5m| / \sqrt{m^2 + 1} = 2$ .
Final Calculation: Squaring both sides to eliminate the absolute value and the square root, we get $(25m^2) / (m^2 + 1) = 4$ .
Error Correction: Multiplying both sides by $(m^2 + 1)$ to clear the denominator, we get $25m^2 = 4m^2 + 4$ .
Error Correction: Multiplying both sides by $m^2 + 1$ to clear the denominator, we get $25m^2 = 4m^2 + 4$ . Subtracting $4m^2$ from both sides, we get $21m^2 = 4$ .
Error Correction: Multiplying both sides by $m^2 + 1$ to clear the denominator, we get $25m^2 = 4m^2 + 4$ . Subtracting $4m^2$ from both sides, we get $21m^2 = 4$ . Dividing both sides by $21$ , we get $m^2 = \frac{4}{21}$ .
Error Correction: Multiplying both sides by $(m^2 + 1)$ to clear the denominator, we get $25m^2 = 4m^2 + 4$ . Subtracting $4m^2$ from both sides, we get $21m^2 = 4$ . Dividing both sides by $21$ , we get $m^2 = \frac{4}{21}$ . Taking the square root of both sides, we get $m = \sqrt{\frac{4}{21}}$ or $m = -\sqrt{\frac{4}{21}}$ . But there's a mistake here; we should have taken the square root of $4/21$ , not $4$ divided by $21$ .