Question

The line $2x+5y=12$ cuts the ellipse $4x^2+5y^2=20$ in $A$ and $B.$ Then the mid point of the chord is

• A. $(2,1)$
• B. $(-2,-1)$
• C. $(-1,-2)$
• D. $(1,2)$

Answer 1

The correct option is D $(1,2)$

$2x+5y=12$
$y=\frac{12-2x}{5}$
$4x^2+5y^2=20$
$\Rightarrow 4x^2+5(\frac{12-2x}{5}{)}^2=20$
$\Rightarrow 24x^2-48x+44=0$
$6x^2-12x+11=0$
$x_1+x_2=\frac{-b}{a}=\frac{12}{6}=2$
$\therefore$ x -coordinate of the midpoint is $\frac{x_1+x_2}{2}=\frac{2}{2}=1$
$\Rightarrow$ y - co.ordinate of the mid-point is $\frac{12-(x_1+x_2)}{5}=\frac{12-2}{5}=2$ ($\because$ it lies on the given line $2x+5y=12$)
$\therefore$ midpoint is $(1,2)$