Find the equation of tangents of the circle
`x^(2) + y^(2)-10 = 0` at the points whose
abscissae are 1.

Answer

To find the equations of the tangents to the circle given by the equation \( x^2 + y^2 - 10 = 0 \) at the points where the abscissae (x-coordinates) are 1, we will follow these steps:

### Step 1: Find the y-coordinates for x = 1
We start by substituting \( x = 1 \) into the circle's equation:

\[ 1^2 + y^2 - 10 = 0 \]

This simplifies to:

\[ 1 + y^2 - 10 = 0 \implies y^2 = 10 - 1 \implies y^2 = 9 \]

Taking the square root gives us:

\[ y = \pm 3 \]

Thus, the points on the circle where \( x = 1 \) are \( (1, 3) \) and \( (1, -3) \).

### Step 2: Differentiate the circle's equation
Next, we differentiate the equation of the circle \( x^2 + y^2 = 10 \) implicitly with respect to \( x \):

\[ \frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(10) \]

This gives:

\[ 2x + 2y \frac{dy}{dx} = 0 \]

Rearranging for \( \frac{dy}{dx} \):

\[ 2y \frac{dy}{dx} = -2x \implies \frac{dy}{dx} = -\frac{x}{y} \]

### Step 3: Find the slope of the tangents at the points
Now, we will calculate the slope of the tangent lines at the points \( (1, 3) \) and \( (1, -3) \).

1. For the point \( (1, 3) \):

\[ \frac{dy}{dx} \bigg|_{(1, 3)} = -\frac{1}{3} \]

2. For the point \( (1, -3) \):

\[ \frac{dy}{dx} \bigg|_{(1, -3)} = -\frac{1}{-3} = \frac{1}{3} \]

### Step 4: Write the equations of the tangents
Using the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \):

1. For the point \( (1, 3) \) with slope \( -\frac{1}{3} \):

\[ y - 3 = -\frac{1}{3}(x - 1) \]

Multiplying through by 3 to eliminate the fraction:

\[ 3(y - 3) = -(x - 1) \implies 3y - 9 = -x + 1 \implies 3y + x = 10 \]

2. For the point \( (1, -3) \) with slope \( \frac{1}{3} \):

\[ y + 3 = \frac{1}{3}(x - 1) \]

Again, multiplying through by 3:

\[ 3(y + 3) = x - 1 \implies 3y + 9 = x - 1 \implies 3y - x = -10 \]

### Final Equations of the Tangents
Thus, the equations of the tangents to the circle at the specified points are:

1. \( 3y + x = 10 \)
2. \( 3y - x = -10 \)