Find an equation for the line tangent to the circle x^2 + y^2 = 25 at the point (3, -4),
Step1 would be to make sure the given point actually lies on the line.
Suppose the given point had been given as (1,6). I could repeat the
following steps and get a totally meaningless result, even know nothing
unbecoming would show up in my solution.
Yes, the point does lie on the circle, so .....
Differentiate implicitly,
x^2 + y^2 = 25
2x + 2y dy/dx = 0
dy/dx = -x/y
so at (3, -4) , dy/dx = slope = -3/-4 = 3/4
equation of tangent:
y + 4 = (3/4)(x - 3)
4y + 16 = 3x - 9
3x - 4y = 25
Can you see a pattern here?
To find an equation for the line tangent to a circle at a given point, you need to know the center of the circle and the radius. In this case, you have the equation of the circle, x^2 + y^2 = 25, which represents a circle with center at the origin (0,0) and radius 5.
First, let's find the slope of the line tangent to the circle at the given point (3, -4). The slope of a tangent line to a circle is equal to the negative reciprocal of the slope of the radius that passes through the point of tangency.
To find the slope of the radius, you need the coordinates of the center of the circle and the point of tangency. The center of the circle is (0, 0), and the point of tangency is (3, -4). The slope of the radius passing through these points is given by:
slope = (y2 - y1) / (x2 - x1)
= (-4 - 0) / (3 - 0)
= -4 / 3
Since the tangent line is perpendicular to the radius at the point of tangency, the slope of the tangent line is the negative reciprocal of the slope of the radius. Therefore, the slope of the tangent line is:
tangent_slope = -1 / (-4 / 3)
= 3/4
Now we have the slope of the tangent line, and we also have a point (3, -4) that lies on the line.
Using the point-slope form of a linear equation:
y - y1 = m(x - x1)
where m is the slope and (x1, y1) are the coordinates of a point on the line, we can substitute the values:
y - (-4) = (3/4)(x - 3)
Simplifying the equation:
y + 4 = (3/4)x - (9/4)
Bringing like terms together:
y = (3/4)x - (9/4) - 4
= (3/4)x - (9/4) - (16/4)
= (3/4)x - (25/4)
Therefore, the equation for the line tangent to the circle x^2 + y^2 = 25 at the point (3, -4) is y = (3/4)x - (25/4).