To solve the problem of finding the equation of the chord of contact from the point (3, 2) to the ellipse given by the equation \( x^2 + 4y^2 = 9 \), and then finding the equation of the straight line joining (3, 2) to the midpoint of this chord, we can follow these steps:
### Step 1: Write the equation of the ellipse in standard form.
The given equation of the ellipse is:
\[
x^2 + 4y^2 = 9
\]
Dividing through by 9, we get:
\[
\frac{x^2}{9} + \frac{y^2}{\frac{9}{4}} = 1
\]
This shows that the semi-major axis \( a = 3 \) and the semi-minor axis \( b = \frac{3}{2} \).
### Step 2: Find the equation of the chord of contact.
The chord of contact from a point \( (x_1, y_1) \) to the ellipse is given by the equation:
\[
\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1
\]
Substituting \( (x_1, y_1) = (3, 2) \), \( a^2 = 9 \), and \( b^2 = \frac{9}{4} \):
\[
\frac{3x}{9} + \frac{2y}{\frac{9}{4}} = 1
\]
This simplifies to:
\[
\frac{x}{3} + \frac{8y}{9} = 1
\]
Multiplying through by 9 to eliminate the denominators:
\[
3x + 8y = 9
\]
This is the equation of the chord of contact.
### Step 3: Find the midpoint of the chord of contact.
Let the midpoint of the chord of contact be \( (h, k) \). The equation of the chord of contact can be expressed as:
\[
hx + 4ky = h^2 + 4k^2
\]
From the previous equation \( 3x + 8y = 9 \), we can equate coefficients:
\[
\frac{3}{h} = \frac{8}{4k} = \frac{9}{h^2 + 4k^2}
\]
### Step 4: Solve for \( h \) and \( k \).
From the first two ratios:
\[
\frac{3}{h} = \frac{2}{k} \implies 3k = 2h \implies h = \frac{3k}{2}
\]
From the second and third ratios:
\[
\frac{2}{k} = \frac{9}{h^2 + 4k^2}
\]
Substituting \( h = \frac{3k}{2} \):
\[
\frac{2}{k} = \frac{9}{\left(\frac{3k}{2}\right)^2 + 4k^2}
\]
Calculating \( \left(\frac{3k}{2}\right)^2 \):
\[
\left(\frac{3k}{2}\right)^2 = \frac{9k^2}{4}
\]
Thus, we have:
\[
\frac{2}{k} = \frac{9}{\frac{9k^2}{4} + 4k^2} = \frac{9}{\frac{9k^2 + 16k^2}{4}} = \frac{36}{25k^2}
\]
Cross multiplying gives:
\[
2 \cdot 25k^2 = 36k \implies 50k^2 - 36k = 0 \implies k(50k - 36) = 0
\]
Thus, \( k = 0 \) or \( k = \frac{36}{50} = \frac{18}{25} \).
Substituting \( k = \frac{18}{25} \) back to find \( h \):
\[
h = \frac{3 \cdot \frac{18}{25}}{2} = \frac{54}{50} = \frac{27}{25}
\]
### Step 5: Find the equation of the line joining (3, 2) to (h, k).
The coordinates of the midpoint are \( \left(\frac{27}{25}, \frac{18}{25}\right) \). The slope of the line joining the points \( (3, 2) \) and \( \left(\frac{27}{25}, \frac{18}{25}\right) \) is:
\[
\text{slope} = \frac{\frac{18}{25} - 2}{\frac{27}{25} - 3} = \frac{\frac{18 - 50}{25}}{\frac{27 - 75}{25}} = \frac{-32/25}{-48/25} = \frac{32}{48} = \frac{2}{3}
\]
Using the point-slope form of the line:
\[
y - 2 = \frac{2}{3}(x - 3)
\]
Simplifying gives:
\[
3y - 6 = 2x - 6 \implies 2x - 3y = 0
\]
### Final Result:
The equation of the chord of contact is \( 3x + 8y = 9 \) and the equation of the line joining \( (3, 2) \) to the midpoint \( \left(\frac{27}{25}, \frac{18}{25}\right) \) is:
\[
2x - 3y = 0
\]
