Question

Let P(1, 1) and Q(3, 2) be given points. the point R on the x-axis such that PR+RQ is minimum is

• A. $(0,\frac{5}{2})$
• B. $(2,0)$
• C. $(3,0)$
• D. $\left(\frac{3}{2},0\right)$

Answer 1

The correct option is A $(0,\frac{5}{2})$

Consider the reflection of point $P$ in the $x-axis$ $P^{\prime }$

$P^{\prime }\equiv (1,-1)$

The $x-axis$ is the $\perp ar$ bisector of $P{P}^{\prime }$

$RP=R{P}^{\prime }$

Adding $RQ$

$RP+RQ=R{P}^{\prime }+RQ$

The sum $(R{P}^{\prime }+RQ)$ is min. when $R$ lies on the line joining $P^{\prime }$ and $Q$.

Now the eqn of the line joining $P^{\prime }(1,-1)$ & $Q(3,2)$

$(y+1)=\frac{2+1}{3-1}(x-1)$

$y=-1+\frac{3x}{2}-\frac{3}{2}$

$y=\frac{3x}{2}-\frac{5}{2}$

The point $R$ lying on $x-axis$ has coordinate zero. So let it have coordinate $(a,0)$ Since $R$ lies on $P^{\prime }Q$;it should satisfy the eqn of $P^{\prime }Q$.

$0=\frac{3a}{2}-\frac{5}{2}$

$a=\frac{5}{2}$

$R(0,\frac{5}{2})$