Q.The line $2x + 3y = 12$ touches the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2$ at the point $(3, 2)$.

The key idea is the Equation of Tangent to an Ellipse. A line touches an ellipse at a point if the point lies on both the line and the ellipse, and the line is the tangent to the ellipse at that point.

1. Verify point on line: Substitute $(3, 2)$ into $2x + 3y = 12$:

   $2(3) + 3(2) = 6 + 6 = 12$. The point $(3, 2)$ lies on the line.

2. Verify point on ellipse: Substitute $(3, 2)$ into $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2$:

   $\dfrac{3^2}{9} + \dfrac{2^2}{4} = \dfrac{9}{9} + \dfrac{4}{4} = 1 + 1 = 2$. The point $(3, 2)$ lies on the ellipse.

3. Derive tangent equation: The equation of the tangent to the ellipse $\dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1$ at $(x_1, y_1)$ is $\dfrac{xx_1}{A^2} + \dfrac{yy_1}{B^2} = 1$.

   The given ellipse is $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2$, which can be written as $\dfrac{x^2}{18} + \dfrac{y^2}{8} = 1$. …

When a line "touches" an ellipse at a specific point, it means two things simultaneously:

1. The given point must lie on the ellipse. If it doesn't, the line cannot touch the ellipse at that point.
2. The given line must be the tangent to the ellipse at that specific point. If the line passes through the point but is not the tangent, it would either intersect the ellipse at two points or not be a tangent at all.

Our strategy will be to check both these conditions.

1. Verify if the point $(3, 2)$ lies on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2$. To do this, we substitute the coordinates $(x, y) = (3, 2)$ into the ellipse equation:

Since the left-hand side equals the right-hand side ($2 = 2$), the point $(3, 2)$ indeed lies on the ellipse. This satisfies the first condition.

2. Find the equation of the tangent to the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2$ at the point $(3, 2)$.

For a general ellipse given by $Ax^2 + By^2 = C$, the equation of the tangent at a point $(x_1, y_1)$ on the ellipse is found by replacing $x^2$ with $xx_1$ and $y^2$ with $yy_1$. This is a standard method for finding tangents to conic sections.

> [!FORMULA]
> The equation of the tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = k$ at a point $(x_1, y_1)$ on the ellipse is $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = k$.

In our case, the ellipse is $\frac{x^2}{9} + \frac{y^2}{4} = 2$, and the point of tangency is $(x_1, y_1) = (3, 2)$.
Applying the formula:

$$ \frac{x}{3} + \frac{y}{2} = 2 $$ …