If a tangent to the ellipse meets major and minor axis at M and N respectively and C is the centre of the ellipse then `(a^(2))/(CM)^(2)+(b^(2))/(CN)^(2)`=

Options

A. (a) 1
B. (b) 2
C. (c) 3
D. (d) 4

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{a^2}{(CM)^2} + \frac{b^2}{(CN)^2} \]

where \(M\) and \(N\) are the points where the tangent to the ellipse meets the major and minor axes, respectively, and \(C\) is the center of the ellipse.

### Step 1: Understand the Geometry of the Ellipse
The standard equation of the ellipse is given by:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. The center \(C\) of the ellipse is at the origin (0, 0).

### Step 2: Equation of the Tangent Line
The equation of the tangent to the ellipse at a point \(P(a \cos \theta, b \sin \theta)\) is given by:

\[ \frac{x \cos \theta}{a} + \frac{y \sin \theta}{b} = 1 \]

### Step 3: Find the Intersection with the Major Axis (X-axis)
To find the point \(M\) where the tangent intersects the major axis (X-axis), set \(y = 0\) in the tangent equation:

\[ \frac{x \cos \theta}{a} = 1 \implies x = \frac{a}{\cos \theta} \]

Thus, the coordinates of point \(M\) are:

\[ M\left(\frac{a}{\cos \theta}, 0\right) \]

### Step 4: Find the Intersection with the Minor Axis (Y-axis)
To find the point \(N\) where the tangent intersects the minor axis (Y-axis), set \(x = 0\) in the tangent equation:

\[ \frac{y \sin \theta}{b} = 1 \implies y = \frac{b}{\sin \theta} \]

Thus, the coordinates of point \(N\) are:

\[ N\left(0, \frac{b}{\sin \theta}\right) \]

### Step 5: Calculate Distances \(CM\) and \(CN\)
The distance \(CM\) from the center \(C(0, 0)\) to point \(M\) is:

\[ CM = \sqrt{\left(\frac{a}{\cos \theta}\right)^2 + 0^2} = \frac{a}{\cos \theta} \]

The distance \(CN\) from the center \(C(0, 0)\) to point \(N\) is:

\[ CN = \sqrt{0^2 + \left(\frac{b}{\sin \theta}\right)^2} = \frac{b}{\sin \theta} \]

### Step 6: Substitute into the Expression
Now substitute \(CM\) and \(CN\) into the given expression:

\[ \frac{a^2}{(CM)^2} + \frac{b^2}{(CN)^2} = \frac{a^2}{\left(\frac{a}{\cos \theta}\right)^2} + \frac{b^2}{\left(\frac{b}{\sin \theta}\right)^2} \]

This simplifies to:

\[ \frac{a^2}{\frac{a^2}{\cos^2 \theta}} + \frac{b^2}{\frac{b^2}{\sin^2 \theta}} = \cos^2 \theta + \sin^2 \theta \]

### Step 7: Use the Pythagorean Identity
Using the identity \(\cos^2 \theta + \sin^2 \theta = 1\):

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

### Final Answer
Thus, we conclude that:

\[ \frac{a^2}{(CM)^2} + \frac{b^2}{(CN)^2} = 1 \]