First slide

Equation of line in Straight lines

Question

A variable line whose slope is -2 cuts x and y-axes respectively at points A and C. If Rhombus ABCD is completed such that the vertex B lies on the line y = x , then the locus of the vertex D is

Difficult

A

$x + y + 2 = 0$
B

$x + y + 3 = 0$
C

$x + y = 0$
D

$x + y - 1 = 0$

Solution

$Let the line be y = - 2 x + c$
$Let B be (α, α) and D be (h, k)$
$A \equiv (\frac{c}{2}, 0) and C \equiv (0, c)$
Comparing midpoints of AC and BD
$\begin{aligned} ∴ h + α = \frac{c}{4} and k + α = \frac{c}{2} \\ Eliminating c, 2 h + 2 α = k + α \end{aligned}$
$or k - 2 h = α - - - - - (1)$
$\begin{aligned} Also A C ⊥ B D \\ \frac{α - k}{α - h} \times (- 2) = - 1 \\ ∴ 2 α - 2 k = α - h \end{aligned}$
$∴ 2 k - h = α - - - - (2)$
$From (1) and (2), k - 2 h = 2 k - h or x + y = 0, which is required locus$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App