Pair of Tangents from a Point to a Circle

Circle Equation: x² + y² = 25
Point: (4, 10)

To find the pair of tangents from a point to a circle, we can use the following steps:

Step 1: Find the equation of the line passing through the center of the circle and the given point.
Step 2: Calculate the distance between the center of the circle and the given point.
Step 3: Find the square of the radius of the circle.
Step 4: Calculate the square of the distance between the center and the given point.
Step 5: Calculate the difference between the square of the radius and the square of the distance.
Step 6: If the difference is negative, there are no real tangents. If the difference is zero, there is one tangent. If the difference is positive, there are two tangents.
Step 7: Find the point(s) of intersection between the line from step 1 and the circle.
Step 8: Calculate the equations of the tangents passing through the intersection point(s) found in step 7.

Let's go through each step in detail:

Step 1: Find the equation of the line passing through the center of the circle and the given point.
The center of the circle is at the origin (0, 0) since the equation of the circle is x² + y² = 25. Therefore, the equation of the line passing through the center and the given point (4, 10) can be found using the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope of the line.
The slope of the line passing through the origin and (4, 10) can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁) = (10 - 0) / (4 - 0) = 10 / 4 = 5/2.
Using the point-slope form, we can substitute the values to find the equation of the line: y - 0 = (5/2)(x - 0) => y = (5/2)x.

Step 2: Calculate the distance between the center of the circle and the given point.
The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by: d = sqrt((x₂ - x₁)² + (y₂ - y₁)²).
In this case, the center of the circle is at (0, 0) and the given point is (4, 10). Substituting the values in the distance formula, we get: d = sqrt((4 - 0)² + (10 - 0)²) = sqrt(16 + 100) = sqrt(116) = 2√29.

Step 3: Find the square of the radius of the circle.
The radius of the circle is the square root of the constant term in the circle equation. In this case, the radius is

Use SS'=T² to get the equation of the pair of tangents.