# Proof of x and y int of a line

• lax1113
In summary, to prove that \sqrt{x} + \sqrt{y} = c, we need to show that the sum of the x and y intercepts of any tangent to the line is equal to a positive constant. To do this, we can use the point-slope form of the tangent line and the correct derivative, which is -\sqrt{x}/\sqrt{y}. By setting y = 0 and x = 0 in the equation and solving for the intercepts, we can show that they add up to be \sqrt{x} + \sqrt{y}.
lax1113

## Homework Statement

prove that $$\sqrt{x}$$ + $$\sqrt{y}$$ = c
Show that the sum of the x and y intercepts of any tangents to the line above = c (some positive constant).

## Homework Equations

y1 - y2 = m(x1 - x2)
dy/dx(for this problem) = -$$\sqrt{y}$$/$$\sqrt{x}$$

## The Attempt at a Solution

So I get the slope, as written above, and put it into a point/slope equation, but where from here? When I try to solve for y = 0 and x = 0 I always have y2 and x2 left, I think I might just be doing something completely wrong, I haven't done something like this for a while. Is this even the right direction? Solve for the y and x int by making the opposite 0 in the equation, and then try to get the results, (the two interecepts) to add up to be $$\sqrt{x}$$ + $$\sqrt{y}$$

First problem, I don't know what "x1", "y1", "x2", and "y2" are since you don't say. You mean, I think, that the equation of the tangent line at $(x_1, y_1)$ is $y- y_1= m(x- x_1). But your real problem is that the derivative of y with respect to x is NOT [itex]-\sqrt{y}/\sqrt{x}$. You have x and y reversed.

= c

In order to prove that the sum of the x and y intercepts of any tangents to the line \sqrt{x} + \sqrt{y} = c is equal to c, we need to use the definition of tangents and the properties of lines.

First, let's define a tangent line as a line that intersects a curve at exactly one point and has the same slope as the curve at that point. In this case, the curve is the line \sqrt{x} + \sqrt{y} = c.

Next, we can use the point-slope form of a line to find the equation of a tangent line to this curve. Since we know that the slope of the curve at any point is given by dy/dx = -\sqrt{y}/\sqrt{x}, we can use the point-slope form to find the equation of the tangent line at any given point (x,y).

So, the equation of the tangent line at (x,y) is y - y = -\sqrt{y}/\sqrt{x}(x - x). Simplifying this, we get y = -\sqrt{y}/\sqrt{x}x + y.

Now, we can use the definition of intercepts to find the x and y intercepts of this tangent line. The x intercept occurs when y = 0, so we can substitute this into our equation to get x = -\sqrt{y}/\sqrt{x}x. Solving for x, we get x = 0.

Similarly, the y intercept occurs when x = 0, so we can substitute this into our equation to get y = -\sqrt{y}/\sqrt{x}(0) + y. Simplifying, we get y = y.

So, the tangent line intersects the x axis at x = 0 and the y axis at y = y. The sum of these intercepts is 0 + y = y.

Now, we can use the original equation \sqrt{x} + \sqrt{y} = c to substitute for y in terms of x. Solving for y, we get y = (c - \sqrt{x})^2.

Substituting this into our equation for the y intercept, we get y = -\sqrt{(c - \sqrt{x})^2}/\sqrt{x}x + (c - \sqrt{x})^2.

Simplifying, we get y = -c

## 1. What is the definition of "Proof of x and y int of a line"?

The proof of x and y intercepts of a line is a mathematical process used to determine the point where a line intersects the x-axis and the y-axis on a coordinate plane.

## 2. How is the x-intercept of a line calculated?

The x-intercept of a line can be calculated by setting the y-coordinate to 0 and solving for the x-coordinate using the equation of the line.

## 3. What is the significance of the x-intercept in a line?

The x-intercept represents the point where the line crosses the x-axis, which is the vertical axis on a coordinate plane. This point is used to determine the slope and to graph the line.

## 4. How is the y-intercept of a line determined?

The y-intercept of a line can be determined by setting the x-coordinate to 0 and solving for the y-coordinate using the equation of the line.

## 5. Why is it important to find the x and y intercepts of a line?

The x and y intercepts provide valuable information about the behavior of a line and can be used to graph the line, find the slope, and solve problems involving the line.

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