First slide

Ellipse

Question

Let AB be a line segment of length 4 units with the point A on the line y = 2x and B on the line y = x. Then the locus of middle point of all such line segment is

Moderate

A

a parabola
B

an ellipse
C

a hyperbola
D

a circle

Solution

Let $B = (α, α)$ and middle point AB is (h,k), Then $A \equiv (2 h - α, 2 k - α)$ lies on y = 2x then
$α = 4 h - 2 k ........ (1) | A B | = 4$
$\Rightarrow {[h - (4 h - 2 k)]}^{2} + {[k - (4 h - 2 k)]}^{2} = 4$ or $25 h^{2} + 13 k^{2} - 36 h k = 4$
Required locus is $25 x^{2} + 13 y^{2} - 36 x y - 4 = 0$
Here, $h^{2} < a b$ and $Δ \neq 0$
$∴$ ellipse

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