Let $ \alpha, \beta, \gamma, \delta \in \mathbb{Z} $ and let $ A (\alpha, \beta) $, $ B (1, 0) $, $ C (\gamma, \delta) $, and $ D (1, 2) $ be the vertices of a parallelogram $ ABCD $. If $ AB = \sqrt{10} $ and the points $ A $ and $ C $ lie on the line $ 3y = 2x + 1 $, then $ 2 (\alpha + \beta + \gamma + \delta) $ is equal to

Question

If eccentricity of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which passes through the point $(3, 4)$ is $\frac{\sqrt{5}}{3}$, then length of latus rectum of ellipse, is :

Answer 1

To solve this problem, we need to determine the value of $ 2 (\alpha + \beta + \gamma + \delta) $ given the parallelogram conditions.

Step 1: Analyze the Line Equation Constraint.

Points $ A(\alpha, \beta) $ and $ C(\gamma, \delta) $ are on the line $ 3y = 2x + 1 $. This yields:

For $ A(\alpha, \beta) $: $3\beta = 2\alpha + 1$.
For $ C(\gamma, \delta) $: $3\delta = 2\gamma + 1$.

Step 2: Apply the Distance Constraint.

The distance $ AB = \sqrt{10} $. Using points $ A(\alpha, \beta) $ and $ B(1, 0) $:

$AB = \sqrt{(\alpha - 1)^2 + (\beta - 0)^2} = \sqrt{10}$

Squaring both sides results in:

$(\alpha - 1)^2 + \beta^2 = 10$

Step 3: Employ Parallelogram Properties.

In a parallelogram, opposite sides are equal and parallel. Thus, $ \overrightarrow{AB} = \overrightarrow{CD} $ and $ \overrightarrow{AD} = \overrightarrow{BC} $. Calculate the vectors:

$\overrightarrow{AB} = (1 - \alpha, -\beta)$
$\overrightarrow{CD} = (1 - \gamma, 2 - \delta)$
Equating components: $1 - \alpha = 1 - \gamma$ and $-\beta = 2 - \delta$.

These simplify to:

$\alpha = \gamma$
$\beta + \delta = 2$

Step 4: Solve the System of Equations.

The derived equations are:

$3\beta = 2\alpha + 1$
$3\delta = 2\gamma + 1$ (which is $3\delta = 2\alpha + 1$ since $\alpha = \gamma$).
$(\alpha - 1)^2 + \beta^2 = 10$
$\beta + \delta = 2$

Substitute equation 4 into equations 1 and 2:

If $\beta = 1$, then $\delta = 1$.

Substitute into a line equation: $3(1) = 2\alpha + 1 \Rightarrow \alpha = 1$.

Step 5: Evaluate the Target Expression.

With $\alpha = 1$, $\beta = 1$, $\gamma = 1$, $\delta = 1$, calculate the expression:

$2(\alpha + \beta + \gamma + \delta) = 2(1 + 1 + 1 + 1) = 8$

Conclusion

The final result is 8.

The Correct Option is D

Solution and Explanation