# Straight Lines - Class 11 - Mathematics

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## Extra Questions - Straight Lines | NCERT | Mathematics | Class 11

The lines joining the points of intersection of the line $x+y=1$ and curve $x^{2}+y^{2}-2y+\lambda=0$ to the origin are perpendicular. Then the value of $\lambda$ will be:

A) $1/2$

B) $-1/2$

C) $1/\sqrt{2}$

D) $0

The correct option is **D (0)**.

Considering the given line $x + y = 1$, we can transform the curve equation $x^2 + y^2 - 2y + \lambda = 0$ into a homogeneous equation by substituting $y = 1 - x$. The transformed equation becomes: $$ x^{2} + (1-x)^{2} - 2(1-x) + \lambda(1)^{2} = 0 \ \Rightarrow x^{2} + (1 - 2x + x^{2}) - 2 + 2x + \lambda = 0 \ \Rightarrow 2x^{2} - 2 + \lambda = 0. $$

**Homogenization:** Convert it by expanding and collecting like terms:
$$
x^2 + y^2 - 2y(x + y) + \lambda(x + y)^2 = 0 \
\Rightarrow x^2 (1 + \lambda) + y^2 (1 - \lambda) - 2xy = 0.
$$

For the lines to be perpendicular (originating from the intersection of the given line and curve and meeting at the origin), the sum $A + B$ in the general form $Ax^2 + By^2 + Cxy = 0$ should be zero:

$$ 1 + \lambda - 1 + \lambda = 0 \Rightarrow 2\lambda = 0 \Rightarrow \lambda = 0. $$

Therefore, **$\boldsymbol{\lambda = 0}$** is the correct value.

A line graph can be:

A. A dotted line

B. A whole unbroken line

C. A zigzag disconnected line

D. None of the above

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