Question

Point $Q$ is symmetric to $P(4,-1)$ with respect to the bisector of the first quadrant. The length of $PQ$ is
A. $3\sqrt{2}$
B. $5\sqrt{2}$
C. $7\sqrt{2}$
D. $9\sqrt{2}$

Answer 1

To solve this problem, first we have to find the symmetric point with the help of the given point and after that we have to find the distance between the given point and the symmetric point by using the distance formula and you will get your required answer.
Complete step-by-step solution:
A line can be defined as a one dimensional geometric shape. It is measured in respect to only one dimension (i.e. length). The points where the line crosses the two axes are called the intercepts. There are basically two intercepts, $x$ intercept and $y$ intercept. The point of intersection of the line with $X$ axis gives the $x$ intercept definition. And the point of intersection of the line with $Y$ axis gives the $y$ intercept definition.
The point where the line or curve crosses the axis of the graph is called an intercept. If the axis is not specified, usually the $Y$ axis is considered.
The standard forms of the equation of a line are: Slope-intercept form, Intercept form, Normal form
Slope-intercept form is the general form of the straight line equation. It is represented as: $y=mx+c$ where $c$ is the intercept and $m$ is the slope, that’s why it is called slope intercept form. The value of $m$ and $c$ are real numbers. The slope of the line is also termed as gradient.
Point slope form is one of the more commonly used forms of a linear equation, and has the following structure: $y-{{y}_{1}}=m(x-{{x}_{1}})$ where $m$ is the slope of the line and $({{x}_{1}},{{y}_{1}})$ is a point on the line. Point slope form is used when one point of the line and the slope are known.
As we are given in the question:
The point $Q$ is symmetric to $P(4,-1)$ with respect to the bisector of the first quadrant.
Equation of the bisector of the first quadrant $y=x$
As, $Q$ is symmetric.
So, $Q$ is at $(-1,4)$
Length of $PQ=\sqrt{{{(4-(-1))}^{2}}+{{(-1-4)}^{2}}}$
$\Rightarrow PQ=\sqrt{{{(5)}^{2}}+{{(-5)}^{2}}}$
$\Rightarrow PQ=\sqrt{25+25}$
$\Rightarrow PQ=\sqrt{50}$
$\Rightarrow PQ=5\sqrt{2}$
The length of the $PQ=5\sqrt{2}$
Hence, the correct option is $D$.

Note: The slope intercept is the most “popular” form of a straight line. Many students find this useful because of its simplicity, because one can easily describe the characteristics of the straight line even without seeing the graph because with the help of this form, slope and intercept can easily be identified.