Equation of Pair of Tangents

When studying conic sections such as parabolas, ellipses, and hyperbolas, the concept of tangents is fundamental. A tangent to a curve at a given point is a straight line that just touches the curve at that point. The equation of a pair of tangents drawn from a point to a conic section is a single algebraic equation that represents both tangents simultaneously.

Pair of Tangents to a Parabola

For a standard parabola with the equation $y^2 = 4ax$, the equation of the tangent at any point $(x_1, y_1)$ on the parabola is given by:

$$ yy_1 = 2a(x + x_1) $$

If we have a point $(x_2, y_2)$ outside the parabola, we can find the pair of tangents from this point to the parabola by using the concept of homogenization. The combined equation of the pair of tangents from $(x_2, y_2)$ to the parabola $y^2 = 4ax$ is given by:

$$ y^2 - 4ax = 0 $$

Homogenizing with the line $y = mx + c$ passing through $(x_2, y_2)$, we get:

$$ y^2 - 4a(mx + c) = 0 $$

Substituting $y = mx + c$ and simplifying, we obtain the equation of the pair of tangents:

$$ c^2 = a(m^2 - 4a) $$

Pair of Tangents to a Circle

For a circle with the equation $x^2 + y^2 = r^2$, the equation of the tangent at any point $(x_1, y_1)$ on the circle is given by:

$$ xx_1 + yy_1 = r^2 $$

The equation of the pair of tangents from an external point $(x_2, y_2)$ can be found using the director circle. The combined equation of the pair of tangents from $(x_2, y_2)$ to the circle $x^2 + y^2 = r^2$ is:

$$ (x_2x + y_2y)^2 = r^2(x^2 + y^2) $$

Differences and Important Points

Here is a table summarizing the differences and important points between the pair of tangents to a parabola and a circle:

Aspect	Parabola	Circle
Standard Equation	$y^2 = 4ax$	$x^2 + y^2 = r^2$
Equation of Tangent	$yy_1 = 2a(x + x_1)$	$xx_1 + yy_1 = r^2$
Point of Contact	$(x_1, y_1)$ on the parabola	$(x_1, y_1)$ on the circle
External Point	$(x_2, y_2)$	$(x_2, y_2)$
Equation of Pair	$c^2 = a(m^2 - 4a)$	$(x_2x + y_2y)^2 = r^2(x^2 + y^2)$
Homogenization	Required	Not required
Director Circle	Not applicable	Used to derive the equation of the pair

Examples

Example 1: Parabola

Find the equation of the pair of tangents to the parabola $y^2 = 8x$ from the point $(1, 4)$.

Solution:

The equation of the parabola is $y^2 = 4ax$, where $a = 2$. The combined equation of the pair of tangents from $(x_2, y_2) = (1, 4)$ is:

$$ c^2 = a(m^2 - 4a) $$

Substituting the values, we get:

$$ 4^2 = 2(m^2 - 8) $$

Solving for $m$, we find the slopes of the tangents. Then, we can write the equations of the tangents using the point-slope form.

Example 2: Circle

Find the equation of the pair of tangents to the circle $x^2 + y^2 = 25$ from the point $(6, 8)$.

Solution:

The equation of the circle is $x^2 + y^2 = r^2$, where $r = 5$. The combined equation of the pair of tangents from $(x_2, y_2) = (6, 8)$ is:

$$ (6x + 8y)^2 = 25(x^2 + y^2) $$

Expanding and simplifying, we get the equation representing the pair of tangents.

Understanding the equation of a pair of tangents is crucial for solving problems related to tangency in conic sections. It is a powerful tool in geometry and is often tested in exams.