Find the equation of the tangent to the circle $x^2+y^2+4 x-10 y=12$ at the point $(3,1)$.
Answer (1)
Find the center and radius of the circle by completing the square: ( x + 2 ) 2 + ( y − 5 ) 2 = 41 .
Calculate the slope of the radius connecting the center ( − 2 , 5 ) and the point ( 3 , 1 ) : m r = 5 − 4 .
Determine the slope of the tangent line as the negative reciprocal of the radius's slope: m t = 4 5 .
Apply the point-slope form to find the equation of the tangent line: 5 x − 4 y − 11 = 0 .
Explanation
Problem Analysis We are given the equation of a circle x 2 + y 2 + 4 x − 10 y = 12 and a point ( 3 , 1 ) on the circle. Our goal is to find the equation of the tangent line to the circle at the given point.
Find the Center and Radius First, we need to find the center and radius of the circle. To do this, we complete the square for both x and y terms in the equation of the circle:
x 2 + 4 x + y 2 − 10 y = 12
To complete the square for x 2 + 4 x , we add ( 4/2 ) 2 = 4 to both sides. To complete the square for y 2 − 10 y , we add ( − 10/2 ) 2 = 25 to both sides.
So, we have: x 2 + 4 x + 4 + y 2 − 10 y + 25 = 12 + 4 + 25 ( x + 2 ) 2 + ( y − 5 ) 2 = 41
From this, we can see that the center of the circle is ( − 2 , 5 ) and the radius is 41 .
Calculate the Slope of the Radius Next, we find the slope of the radius connecting the center ( − 2 , 5 ) and the point ( 3 , 1 ) on the circle. The slope m r is given by:
m r = 3 − ( − 2 ) 1 − 5 = 5 − 4
Determine the Slope of the Tangent Since the tangent line is perpendicular to the radius at the point of tangency, the slope of the tangent line m t is the negative reciprocal of the slope of the radius:
m t = − m r 1 = − − 5 4 1 = 4 5
Apply the Point-Slope Form Now, we use the point-slope form of a line to find the equation of the tangent line. The point-slope form is given by:
y − y 1 = m t ( x − x 1 )
where ( x 1 , y 1 ) is the point ( 3 , 1 ) and m t is the slope 4 5 . Substituting these values, we get:
y − 1 = 4 5 ( x − 3 )
Simplify the Equation Finally, we simplify the equation to get the equation of the tangent line in the form A x + B y + C = 0 . Multiply both sides by 4 to get:
4 ( y − 1 ) = 5 ( x − 3 ) 4 y − 4 = 5 x − 15
Rearranging the terms, we get:
5 x − 4 y − 15 + 4 = 0 5 x − 4 y − 11 = 0
So, the equation of the tangent line is 5 x − 4 y − 11 = 0 .
Final Answer The equation of the tangent to the circle x 2 + y 2 + 4 x − 10 y = 12 at the point ( 3 , 1 ) is 5 x − 4 y − 11 = 0 .
Examples
Understanding tangent lines is crucial in various fields. For instance, in physics, when analyzing the motion of an object along a curved path, the tangent line at a specific point gives the direction of the object's velocity at that instant. Similarly, in engineering, tangent lines are used to design smooth transitions in roads and railways, ensuring a comfortable ride. This concept also extends to economics, where tangent lines can represent marginal cost or revenue, helping businesses make informed decisions about production and pricing.
