The equations of the tangents to the ellipse `3x^(2)+y^(2)=3` making equal intercepts on the axes are

Options

A. `y=pm x pm 2`
B. `y=pm x pm 4`
C. `y=pm x pm sqrt30`
D. `y=pm x pm sqrt35`

Answer

To find the equations of the tangents to the ellipse \(3x^2 + y^2 = 3\) that make equal intercepts on the axes, we can follow these steps:

### Step 1: Rewrite the equation of the ellipse in standard form.
The given equation of the ellipse is:
\[ 3x^2 + y^2 = 3 \]
Dividing the entire equation by 3, we get:
\[ \frac{x^2}{1} + \frac{y^2}{3} = 1 \]
This is the standard form of the ellipse.

**Hint:** To convert the equation of the ellipse into standard form, divide by the constant on the right side.

### Step 2: Understand the condition for equal intercepts.
The tangents we are looking for make equal intercepts on the axes. If a line makes equal intercepts \(a\) on both axes, its equation can be expressed as:
\[ x + y = a \quad \text{or} \quad x - y = a \]
This means the slope \(m\) of the line can be either \(1\) or \(-1\).

**Hint:** The slope of the line is crucial for determining the form of the tangent equation.

### Step 3: Write the equation of the tangent line.
Assuming the slope \(m = -1\) for the tangent line, we can write the equation of the tangent line as:
\[ y = -x + a \]

**Hint:** Use the slope-intercept form of a line to express the tangent.

### Step 4: Use the tangent line equation in the ellipse equation.
The general equation of a tangent to the ellipse in slope form is given by:
\[ y = mx + \sqrt{a^2 m^2 + b^2} \]
where \(a^2 = 1\) and \(b^2 = 3\) for our ellipse.

Substituting \(m = -1\):
\[ y = -x + \sqrt{1 \cdot (-1)^2 + 3} = -x + \sqrt{1 + 3} = -x + 2 \]

**Hint:** Substitute the values of \(a\) and \(b\) into the tangent line equation to find the intercept.

### Step 5: Find the other tangent line.
Similarly, for \(m = 1\):
\[ y = x + \sqrt{1 \cdot 1^2 + 3} = x + \sqrt{1 + 3} = x + 2 \]

**Hint:** Repeat the process for the positive slope to find the second tangent.

### Step 6: Combine the results.
Thus, the equations of the tangents that make equal intercepts on the axes are:
\[ y = x + 2 \quad \text{and} \quad y = -x + 2 \]

**Hint:** Combine both results to express the final answer.

### Final Answer:
The equations of the tangents to the ellipse \(3x^2 + y^2 = 3\) making equal intercepts on the axes are:
\[ y = x + 2 \quad \text{and} \quad y = -x + 2 \]