Condition of Tangency (Cartesian Form)

In geometry, the condition of tangency refers to the situation where a line just touches a curve or a surface at a single point. This concept is particularly important in the study of conic sections (like circles, ellipses, parabolas, and hyperbolas) and other surfaces in three-dimensional geometry. In the Cartesian coordinate system, we can express the condition of tangency using algebraic equations.

Tangency to a Circle

For a circle with center at $(h, k)$ and radius $r$, the equation is given by:

$$(x - h)^2 + (y - k)^2 = r^2$$

A line with the equation $y = mx + c$ is tangent to this circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle. The condition for tangency is:

$$|c - k - mh| = r \sqrt{1 + m^2}$$

Tangency to a Parabola

For a parabola with the equation $y^2 = 4ax$, a line with the equation $y = mx + c$ is tangent to the parabola if:

$$c = a/m$$

Tangency to an Ellipse

For an ellipse with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, a line with the equation $y = mx + c$ is tangent to the ellipse if:

$$c^2 = a^2m^2 + b^2$$

Tangency to a Hyperbola

For a hyperbola with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, a line with the equation $y = mx + c$ is tangent to the hyperbola if:

$$c^2 = a^2m^2 - b^2$$

Differences and Important Points

Here is a table summarizing the differences and important points for the condition of tangency for different conic sections:

Conic Section	Equation of Conic	Equation of Line	Condition of Tangency
Circle	$(x - h)^2 + (y - k)^2 = r^2$	$y = mx + c$	$
Parabola	$y^2 = 4ax$	$y = mx + c$	$c = a/m$
Ellipse	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$y = mx + c$	$c^2 = a^2m^2 + b^2$
Hyperbola	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$y = mx + c$	$c^2 = a^2m^2 - b^2$

Examples

Example 1: Tangency to a Circle

Given a circle with center at $(0, 0)$ and radius $5$, and a line with the equation $y = 3x + c$, find the value of $c$ for which the line is tangent to the circle.

Using the condition of tangency for a circle:

$$|c| = 5 \sqrt{1 + 3^2}$$ $$|c| = 5 \sqrt{10}$$

Therefore, the line is tangent to the circle if $c = \pm 5\sqrt{10}$.

Example 2: Tangency to a Parabola

Given a parabola with the equation $y^2 = 8x$, and a line with the equation $y = 2x + c$, find the value of $c$ for which the line is tangent to the parabola.

Using the condition of tangency for a parabola:

$$c = \frac{a}{m}$$ $$c = \frac{8}{2}$$ $$c = 4$$

Therefore, the line is tangent to the parabola if $c = 4$.

Example 3: Tangency to an Ellipse

Given an ellipse with the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and a line with the equation $y = \frac{1}{2}x + c$, find the value of $c$ for which the line is tangent to the ellipse.

Using the condition of tangency for an ellipse:

$$c^2 = 16\left(\frac{1}{2}\right)^2 + 9$$ $$c^2 = 4 + 9$$ $$c^2 = 13$$

Therefore, the line is tangent to the ellipse if $c = \pm \sqrt{13}$.

By understanding the conditions of tangency for different conic sections and surfaces, we can solve various problems related to tangents in Cartesian geometry. These conditions are essential for analyzing geometric figures and their properties, and they are widely used in mathematics, physics, engineering, and other fields.