#### To determine

**To show:** The sum of the *x*-intercepts and *y*-intercepts of any tangent line to the curve x+y=c is equal to constant *c.*

#### Explanation

**Given:**

The equation of the curve is x+y=c.

**Derivative rules:**

*Chain rule*

If y=f(u) and u=g(x) are both differentiable function, then dydx=dydu⋅dudx.

**Formula used:**

The equation of the tangent line at (x1,y1) is, y−y1=m(x−x1) ,where, *m* is the slope of the tangent line at (x1,y1) and m=dydx|x=x1,y=y1

**Proof:**

Obtain the slope of the tangent line to the equation x+y=c as follows.

Differentiate implicitly with respect to *x* on both sides of the above equation,

ddx(x+y)=ddx(c)ddx(x)+ddx(y)=0ddx(x12)+ddx(y12)=0

Apply the chain rule and simplify the terms,

(12x12−1)−[ddy(y12)dydx]=012x−12+(12y12−1)dydx=0(y−12)dydx=−x−12(1y12)dydx=−1x12

Multiply the above equation by y12 on both sides,

dydx=−y12x12dydx=−yx

Thus, the derivative of the equation x+y=c is dydx=−yx.

That is, the slope of tangent to the equation is dydx=−yx.

Consider the point (a,b) on graph of the curve x+y=c.

Therefore, the slope of the curve at the point (a,b) is m=−ba.

Substitute (a,b) for (x1,y1) and slope m=−ba in the above mentioned formula as,

y−b=−ba(x−a)ay−ba=−bx+abbx+ay=ab+ba

Therefore, the equation of the tangent line to the curve at (a,b) is bx+ay=ab+ba.

Obtain the *x*-intercept and *y-*intercept of the tangent line.

Substitute y=0 in the equation of the tangent line bx+ay=ab+ba to the curve at (a,b),

bx+a(0)=ab+babx=ab+bax=ab+babx=a+ab

Thus, the *x*-intercept is x=a+ab.

Similarly, substitute x=0 in the equation of the tangent line bx+ay=ab+ba to the curve at (a,b),

b(0)+ay=ab+baby=ab+bay=ab+baay=b+ab

Thus, the *y*-intercept is y=b+ab.

Thus, the *x*-intercept and *y-*intercept of the tangent line is x=a+ab and y=b+ab,respectively.

Obtain the sum of the *x*-intercept and *y-*intercept of the tangent line as follows.

a+ab+b+ab=a+b+2ab=(a)2+(b)2+2ab=(a+b)2=c [∵a+b=c]

Therefore, the sum of the *x*-intercept and *y-*intercept of the tangent is a constant *c*.

Hence, the required proof is obtained.