The equation of a circle is x2 + y2 + 4x − 2y − 20 = 0. Find:(a) the length of the tangent to this circle from the point (5, 2)

To find the length of the tangent to the circle from the point (5,2), we first need to find the center and radius of the circle.

The general equation of a circle is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle and r is the radius.

The given equation is x² + y² + 4x - 2y - 20 = 0. We can rewrite this in the form of the general equation by completing the square:

(x² + 4x) + (y² - 2y) = 20

(x² + 4x + 4) - 4 + (y² - 2y + 1) - 1 = 20

(x + 2)² + (y - 1)² = 25

So, the center of the circle is (-2,1) and the radius is √25 = 5.

Next, we find the distance from the point (5,2) to the center of the circle. This is given by the formula √[(x₂ - x₁)² + (y₂ - y₁)²]:

d = √[(5 - (-2))² + (2 - 1)²] = √[49 + 1] = √50

The length of the tangent line from the point to the circle is given by the Pythagorean theorem: √(d² - r²):

Length of tangent = √(50 - 25) = √25 = 5

So, the length of the tangent to the circle from the point (5,2) is 5 units.

To find the length of the tangent to the circle from the point (5,2), we first need to find the center and radius of the circle.

The general equation of a circle is (x-h)² + (y-k)² = r², where (h,k) is the center of the circle and r is the radius.

The given equation is x² + y² + 4x - 2y - 20 = 0. We can rewrite this in the form of the general equation by completing the square:

(x² + 4x) + (y² - 2y) = 20 (x² + 4x + 4) - 4 + (y² - 2y + 1) - 1 = 20 (x + 2)² + (y - 1)² = 25

So, the center of the circle is (-2,1) and the radius is √25 = 5.

Next, we find the distance from the point (5,2) to the center of the circle using the distance formula:

d = √[(5 - (-2))² + (2 - 1)²] = √[49 + 1] = √50

The length of the tangent is then found by using the Pythagorean theorem, where the hypotenuse is the distance from the point to the center of the circle, one leg is the radius of the circle, and the other leg is the length of the tangent:

t² = d² - r² t = √[d² - r²] t = √[50 - 25] t = √25 t = 5

So, the length of the tangent to the circle from the point (5,2) is 5 units.