The line `y=x+5` touches

Options

A. the parabola `y^(2)=20x`
B. the ellipse `9x^(2)+16y^(2)=144`
C. the hyperbola `(x^(2))/(29)-(y^(2))/(4)=1`
D. the circle `x^(2)+y^(2)=25`

Answer

To determine which conic sections the line \( y = x + 5 \) touches, we will analyze each of the given curves step by step.

### Step 1: Identify the line's slope and intercept
The equation of the line is given as:
\[ y = x + 5 \]
From this, we can identify:
- Slope \( m = 1 \)
- Intercept \( c = 5 \)

### Step 2: Analyze the first curve (Parabola)
The first curve is given by:
\[ y^2 = 20x \]
We can rewrite this in the standard form of a parabola:
\[ y^2 = 4ax \quad \text{where } 4a = 20 \implies a = 5 \]
The slope form of the tangent to a parabola is given by:
\[ c = \frac{a}{m} \]
Substituting the values:
\[ c = \frac{5}{1} = 5 \]
Since this matches the intercept of the line, the line touches the parabola.

### Step 3: Analyze the second curve (Ellipse)
The second curve is:
\[ 9x^2 + 16y^2 = 144 \]
Dividing through by 144 gives:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
Here, \( a^2 = 16 \) and \( b^2 = 9 \). The slope form of the tangent to an ellipse is given by:
\[ c = \sqrt{a^2 m^2 + b^2} \]
Substituting the values:
\[ c = \sqrt{16(1^2) + 9} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Since this also matches the intercept of the line, the line touches the ellipse.

### Step 4: Analyze the third curve (Hyperbola)
The third curve is:
\[ \frac{x^2}{25} - \frac{y^2}{4} = 1 \]
Here, \( a^2 = 25 \) and \( b^2 = 4 \). The slope form of the tangent to a hyperbola is given by:
\[ c = \sqrt{a^2 - b^2 m^2} \]
Substituting the values:
\[ c = \sqrt{25 - 4(1^2)} = \sqrt{25 - 4} = \sqrt{21} \]
Since \( \sqrt{21} \) is not equal to 5, the line does not touch the hyperbola.

### Step 5: Analyze the fourth curve (Circle)
The fourth curve is:
\[ x^2 + y^2 = 25 \]
This can be rewritten as:
\[ x^2 + y^2 = a^2 \quad \text{where } a^2 = 25 \implies a = 5 \]
The slope form of the tangent to a circle is given by:
\[ c = \sqrt{a^2 + m^2} \]
Substituting the values:
\[ c = \sqrt{25 + 1} = \sqrt{26} \]
Since \( \sqrt{26} \) is not equal to 5, the line does not touch the circle.

### Conclusion
The line \( y = x + 5 \) touches:
- The parabola \( y^2 = 20x \)
- The ellipse \( 9x^2 + 16y^2 = 144 \)
- The hyperbola \( \frac{x^2}{25} - \frac{y^2}{4} = 1 \) (does not touch)
- The circle \( x^2 + y^2 = 25 \) (does not touch)

Thus, the line touches the parabola and the ellipse but not the hyperbola or the circle.

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