Easy tangent problem

Nothing to do, just substitute $x = 3$ and $y = 1$ in $y = mx + b$ . So, $3 = m×1 + b\\ m + b = \color{#69047E}{\boxed{3}}$

The equation ${x}^{2}+{y}^{2}=10$ is the equation of a circle as it is in the form ${(x-a)}^{2}+{(y-b)}^{2}={r}^{2}$ . The centre of the circle is at ${(0,0)}$ as ${a}$ and ${b}$ are 0.

Since a tangent of a circle is perpendicular to its radius at that point, if we find the gradient of the line from ${(0,0)}$ to point P ${(1,3)}$ , and then find it's negative reciprocal (using the formula $M_{1}\times M_{2}={-1}$ ) we can find the gradient of the tangent.

3/1 = 3 = Gradient of radius to point P

${3}\times M_{2}={-1}$ --> $M_{2}=\frac{-1}{3}$ --> -1/3 = Gradient of tangent at point P = m

Now that we have found m, which is -1/3, we need to find b. ${y=-\frac{1}{3}x+b}$

Using point P, 3 = -1/3(1)+b --> 3+1/3 = b --> 10/3 = b

- ${\frac{1}{3} + \frac{10}{3}}$ = ${\frac{9}{3}}$ = 3

m + b = ${\boxed{3}}$

As the point $P(1,3)$ lies on the curve and also the line $y=mx+b$ . Put the point $P$ on the given line we get, $3=m*1+b$ or, $m+b=\boxed{3}$ .

Hello,

2x + 2yy' = 0 at P(1,3),

m = y' = -2x / 2y = - x / y = -1 / 3 , substituting x=1, y=3,

y = -1 / 3 x + b , as x=1 , y=3,

b = 3 + 1/3 = 10/3,

then, just m + b , you will get 3.00000000....

thanks.....