Radius & Tangent

• Perpendicular Gradients: In coordinate geometry, if two lines are perpendicular, the product of their gradients is $-1$. If the gradient of the radius is $m_r$, then the gradient of the tangent $m_t$ must satisfy the equation $m_r \cdot m_t = -1$.

• Gradient Calculation: The gradient of the radius is found using the coordinates of the center $(a, b)$ and the point of tangency $(x_1, y_1)$. The formula is $m_r = \frac{y_1 - b}{x_1 - a}$, representing the vertical change over the horizontal change between these two points.

• Negative Reciprocal: To find the tangent's gradient, we take the negative reciprocal of the radius's gradient. This means $m_t = -\frac{1}{m_r} = -\frac{x_1 - a}{y_1 - b}$, which ensures the line is oriented at exactly $90^{\circ}$ to the radius.

Finding the Equation of a Tangent

• Step 1: Identify the Center: Determine the coordinates of the circle's center $(a, b)$ from the circle equation $(x-a)^2 + (y-b)^2 = r^2$. If the equation is in general form, you must first complete the square to find these coordinates.

• Step 2: Calculate Radius Gradient: Use the center and the given point of tangency $P(x_1, y_1)$ to calculate the gradient $m_r$. Ensure you subtract the coordinates in a consistent order to avoid sign errors.

• Step 3: Determine Tangent Gradient: Calculate the negative reciprocal of $m_r$ to find $m_t$. If $m_r$ is zero (horizontal radius), the tangent is a vertical line ($x = x_1$); if $m_r$ is undefined (vertical radius), the tangent is a horizontal line ($y = y_1$).

• Step 4: Formulate the Line Equation: Substitute $m_t$ and the point $P(x_1, y_1)$ into the point-slope formula: $y - y_1 = m_t(x - x_1)$. Expand and rearrange this into the required format, such as $y = mx + c$ or $ax + by + c = 0$.

• Tangent vs. Normal: While the tangent is perpendicular to the radius, the normal line is the line that contains the radius itself. The normal passes through both the center and the point of tangency, meaning it has the same gradient as the radius.

• Tangent vs. Secant: A tangent touches the circle at exactly one point, whereas a secant is a line that intersects the circle at two distinct points. Algebraically, a tangent results in a single repeated root when solved simultaneously with the circle equation, while a secant results in two distinct roots.

• Verify the Point: Always check if the given point $(x_1, y_1)$ actually lies on the circle by substituting it into the circle's equation. If the point does not satisfy the equation, it is not a point of tangency, and the standard method will not apply.

• Gradient Signs: A common mistake is forgetting the negative sign when finding the perpendicular gradient. If your radius has a positive slope, your tangent MUST have a negative slope (and vice versa), unless one is horizontal/vertical.

• Exact Values: Exams often require answers in exact form. Keep gradients as fractions rather than decimals to maintain precision throughout the calculation, especially when dealing with square roots from the radius calculation.

• Sanity Check: Visualize the circle and the point. If the point is at the 'top' of the circle, the tangent should be roughly horizontal. If the calculation results in a steep vertical gradient, you may have swapped your $x$ and $y$ values.

• Using the Center in the Line Equation: Students often correctly find the tangent gradient but then accidentally use the circle's center $(a, b)$ in the $y - y_1 = m(x - x_1)$ formula. The tangent does NOT pass through the center; it only passes through the point on the circumference.

• Confusing Radius and Diameter: When calculating the gradient, ensure you are using the center and a point on the edge. Using two points on the circumference (a chord) will give a different gradient that is not perpendicular to the tangent at either point.

• Reciprocal Errors: When the radius gradient is a fraction like $\frac{2}{3}$, the tangent gradient is $-\frac{3}{2}$. A common error is to only flip the fraction or only change the sign, rather than doing both.