A normal to the hyperbola `(x^2)/4-(y^2)/1=1` has equal intercepts on the positive x- and y-axis. If this normal touches the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` , then `a^2+b^2` is equal to 5 (b) 25 (c) 16 (d) none of these

Options

A. 5
B. 25
C. 16
D. none of these

Explanation

To solve the problem step by step, we need to find the value of \( a^2 + b^2 \) given the conditions of the hyperbola and the ellipse.

### Step 1: Understand the Hyperbola
The hyperbola given is:
\[ \frac{x^2}{4} - \frac{y^2}{1} = 1 \]
This can be rewritten as:
\[ x^2 - 4y^2 = 4 \]

### Step 2: Equation of the Normal
The equation of the normal to the hyperbola at a point \( (x_0, y_0) \) is given by:
\[ y - y_0 = -\frac{b^2}{a^2}(x - x_0) \]
For our hyperbola, \( a^2 = 4 \) and \( b^2 = 1 \). Thus, the slope of the normal is:
\[ -\frac{1}{4}(x_0 - 2) = -\frac{1}{4}(x_0 - 2) \]

### Step 3: Equal Intercepts Condition
The normal has equal intercepts on the positive x- and y-axis. If the intercepts are equal, we can denote them as \( c \). The equation of the normal can be expressed as:
\[ x + y = c \]
This implies that the intercepts on the axes are \( c \) each.

### Step 4: Finding the Point on the Hyperbola
To find the point \( (x_0, y_0) \) on the hyperbola, we can use the condition of equal intercepts. The x-intercept is \( c \) and the y-intercept is also \( c \). Thus, we can set:
\[ x_0 = c, \quad y_0 = c \]
Substituting into the hyperbola equation:
\[ \frac{c^2}{4} - \frac{c^2}{1} = 1 \]
This simplifies to:
\[ \frac{c^2}{4} - c^2 = 1 \implies -\frac{3c^2}{4} = 1 \implies c^2 = -\frac{4}{3} \]
This is not possible, indicating we need to look at the slope condition instead.

### Step 5: Using the Slope of the Normal
The slope of the normal is given by:
\[ -\frac{b^2}{a^2} \cdot \frac{1}{\sqrt{1 - \frac{y_0^2}{b^2}}} \]
Setting the slope equal to the negative reciprocal of the slope of the line \( y = -x + c \):
\[ -\frac{1}{4} = -\frac{1}{\sqrt{3}} \]
This gives us:
\[ \sin \theta = \frac{1}{2} \implies \theta = \frac{\pi}{6} \]

### Step 6: Finding the Intercept
The intercept can be calculated as:
\[ c = \frac{5}{\sqrt{3}} \]

### Step 7: Condition for the Ellipse
The normal touches the ellipse:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Using the intercepts:
\[ \left(\frac{5}{\sqrt{3}}\right)^2 = a^2 + b^2 \]
This leads to:
\[ \frac{25}{3} = a^2 + b^2 \]

### Final Step: Conclusion
Thus, the value of \( a^2 + b^2 \) is:
\[ \frac{25}{3} \]
Since this does not match any of the options given, the answer is:
\[ \text{(d) none of these} \]