Find the equation of the line tangent to the curve y=(x^2+3)^1/2 that is perpendicular to the line 2x-y+7=0
you want the tangent to have slope -1/2
y' = x/√(x^2+3)
so, where is that = -1/2 ?
Get the point and use the point-slope form of the line.
To find the equation of the line tangent to the curve y = (x^2+3)^(1/2) that is perpendicular to the line 2x - y + 7 = 0, we need to follow these steps:
Step 1: Find the derivative of the given curve.
Step 2: Use the derivative to find the slope of the tangent line.
Step 3: Find the slope of the line perpendicular to the given line.
Step 4: Use the slope of the tangent line and the slope of the perpendicular line to find the slope of the desired tangent line.
Step 5: Use the slope and the given point of tangency to find the equation of the tangent line.
Let's go through these steps in detail:
Step 1: Find the derivative of y = (x^2+3)^(1/2)
To find the derivative, we can use the power rule:
dy/dx = (1/2)((x^2+3)^(-1/2))(2x)
Simplifying this expression, we get:
dy/dx = x / (x^2 + 3)^(1/2)
Step 2: Find the slope of the tangent line
The slope of the tangent line is equal to the derivative of the curve evaluated at the point of tangency. Let's denote the point of tangency as (a, b). So, b = (a^2+3)^(1/2).
Substituting the value of b into the derivative expression, we get:
slope of tangent line = a / (a^2 + 3)^(1/2)
Step 3: Find the slope of the line perpendicular to the given line
The given line has the form 2x - y + 7 = 0.
The slope of this line can be determined by rearranging the equation into slope-intercept form (y = mx + b), where m represents the slope.
Rearranging the given line equation, we get:
y = 2x + 7
Comparing this equation to y = mx + b, we see that the slope of the given line is 2.
The slope of a line perpendicular to the given line is the negative reciprocal of this slope. So, the slope of the perpendicular line is -1/2.
Step 4: Find the slope of the desired tangent line
Since the desired tangent line is perpendicular to the given line, its slope will be the negative reciprocal of the slope of the given line. Therefore, the slope of the desired tangent line is 2.
Step 5: Use the slope and the given point of tangency to find the equation of the tangent line
We have the slope of the tangent line as 2, and the point of tangency as (a, b) where b = (a^2+3)^(1/2).
The equation of a line with slope m and passing through a point (x1, y1) is given by:
y - y1 = m(x - x1)
Substituting the values, we get:
y - (a^2+3)^(1/2) = 2(x - a)
This is the equation of the tangent line to the curve y = (x^2+3)^(1/2) that is perpendicular to the line 2x - y + 7 = 0.