Question

The length of the chord of the ellipse $\frac{x^2}{4} + y^2 = 1$ formed on the line $y = x+1$ is

• A. $2\sqrt{2}$
• B. $4\sqrt{2}/5$
• C. $4\sqrt{2}$
• D. $8\sqrt{2}/5$

Hint:

To find the length of a chord intercepted by a conic section on a line, solve the two equations simultaneously to find the coordinates of the points of intersection. Then, use the distance formula between these two points.

Answer 1

The Correct Option is D

To find the length of the chord, we first need to find the points of intersection of the ellipse and the line.
Ellipse: $\frac{x^2}{4} + y^2 = 1 \implies x^2 + 4y^2 = 4$.
Line: $y = x+1$.
Substitute the expression for $y$ from the line into the ellipse equation:
$x^2 + 4(x+1)^2 = 4$.
$x^2 + 4(x^2+2x+1) = 4$.
$x^2 + 4x^2+8x+4 = 4$.
$5x^2 + 8x = 0$.
$x(5x+8) = 0$.
This gives two x-coordinates for the intersection points: $x_1=0$ and $x_2 = -8/5$.
Now find the corresponding y-coordinates using $y=x+1$:
If $x_1=0$, then $y_1 = 0+1=1$. So, point A is $(0,1)$.
If $x_2=-8/5$, then $y_2 = -8/5 + 1 = -3/5$. So, point B is $(-8/5, -3/5)$.
The length of the chord is the distance between points A and B.
Length = $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
Length = $\sqrt{(-\frac{8}{5}-0)^2 + (-\frac{3}{5}-1)^2} = \sqrt{(-\frac{8}{5})^2 + (-\frac{8}{5})^2}$.
Length = $\sqrt{\frac{64}{25} + \frac{64}{25}} = \sqrt{2 \times \frac{64}{25}} = \frac{8\sqrt{2}}{5}$.