Question Statement

Show that the tangent line at any point $P$ of the circle is always perpendicular to the radial line through point $P$.

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Background and Explanation

To prove that two lines are perpendicular, we must show that the product of their slopes is equal to $-1$. This solution utilizes implicit differentiation to find the slope of the tangent line and the standard slope formula to find the slope of the radius (the radial line).

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Solution

Consider a circle with its center at the origin $(0,0)$ and radius $r$. The equation of the circle is:

$(x-0{)}^2+(y-0{)}^2=r^2$

or

$x^2+y^2=r^2$

Take any point $P(x_1,y_1)$ on the circle.

Step 1: Find the slope of the tangent line

To find the slope of the tangent line, we differentiate equation (i) with respect to $x$ using implicit differentiation:

$\frac{d}{dx}(x^2)+\frac{d}{dx}(y^2)=\frac{d}{dx}(r^2)$

$2x+2y\frac{dy}{dx}=0$

Solving for $\frac{dy}{dx}$:

$2y\frac{dy}{dx}=-2x$

$\frac{dy}{dx}=-\frac{x}{y}$

At the specific point $P(x_1,y_1)$, the slope of the tangent line (let's call it $m_1$) is:

$m_1=-\frac{x_1}{y_1}$

Step 2: Find the slope of the radial line

The radial line (or segment $\overline{CP}$) connects the center $C(0,0)$ to the point $P(x_1,y_1)$. Using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$, the slope of the radial segment (let's call it $m_2$) is:

$m_2=\frac{y_1-0}{x_1-0}=\frac{y_1}{x_1}$

Step 3: Prove perpendicularity

Two lines are perpendicular if the product of their slopes is $-1$. We multiply the slope of the tangent line by the slope of the radial segment:

$(\text{Slope of tangent line})\times (\text{Slope of }\overline{CP})=m_1\cdot m_2$

$\left(-\frac{x_1}{y_1}\right)\left(\frac{y_1}{x_1}\right)=-1$

Since the product of the slopes is $-1$, this shows that the tangent at point $P$ is perpendicular to the radial segment through $P$.

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Key Formulas or Methods Used

• Equation of a circle (centered at origin): $x^2+y^2=r^2$
• Implicit Differentiation: Used to find the derivative of $y$ with respect to $x$ when $y$ is not isolated.
• Slope Formula: $m=\frac{y_2-y_1}{x_2-x_1}$
• Condition for Perpendicular Lines: $m_1\cdot m_2=-1$

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Summary of Steps

1. Set up the equation of the circle $x^2+y^2=r^2$.
2. Differentiate the equation implicitly to find the expression for the slope of the tangent line: $\frac{dy}{dx}=-\frac{x}{y}$.
3. Substitute the coordinates of point $P(x_1,y_1)$ to find the specific slope of the tangent at that point.
4. Calculate the slope of the radius connecting $(0,0)$ to $(x_1,y_1)$.
5. Multiply the two slopes together to verify that their product is $-1$, confirming they are perpendicular.