Question

Assertion :Reflection of the point $(-3,2)$ in the line $x+y=0$ is $(-2,3)$ Reason: The reflection of a point $P(\alpha ,\beta )$ in the line $ax+by+c=0$ is the point $Q({\alpha }^{\prime },{\beta }^{\prime })$, if $\left(\frac{\alpha +{\alpha }^{\prime }}{2},\frac{\beta +{\beta }^{\prime }}{2}\right)$ lies on the line.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

The reflection of a point $P(\alpha ,\beta )$ in the line $ax+by+c=0$ is the point $Q^{\prime }({\alpha }^{\prime },{\beta }^{\prime })$ such that the mid-point of $PQ$i.e $\left(\frac{\alpha +{\alpha }^{\prime }}{2},\frac{\beta +{\beta }^{\prime }}{2}\right)$ lies on the line $ax+by+c=0$ and $PQ$ is perpendicular to the line $ax+by+c=0$.
Also, the coordinates of $Q$ are given by
$\frac{{\alpha }^{\prime }-\alpha }{a}=\frac{{\beta }^{\prime }-\beta }{b}=-2\left(\frac{a\alpha +b\beta +c}{a^2+b^2}\right)$
So, statement-2(Reason) is not true.
The reflection $(-3,2)$ in the line $x+y=0$ has the coordinates $(x,y)$ given by
$\frac{x+3}{1}=\frac{y-2}{1}=-\frac{2(-3+2)}{1^2+1^2}$
$\Rightarrow x+3=y-2=1\Rightarrow x=-2$ and $y=3$.
So, statement-1 (Assertion) is true.