Question

If the straight line, $2x-3y+17=0$ is perpendicular to the line passing through the points $\left( 7,17 \right)$ and $\left( 15,\beta \right)$ , then $\beta $ equals:-
A. $-5$
B. $-\dfrac{35}{3}$
C. $\dfrac{35}{3}$
D. $5$

Answer 1

Hint: The given problem is related to slope of a line. Try to recall the ways to find the slope if the equation of a line is given. Then use the relation between the slopes of perpendicular lines to find the slope of the line perpendicular to the given line. Then use the formula of slope of line passing through two points to find the value of $\beta $ .

Complete answer:
The given equation of the line is $2x-3y+17=0$. We will rearrange the equation to get it in the form of $y=mx+c$ . We have $2x-3y+17=0$.
$\Rightarrow 3y=2x+17$
$\Rightarrow y=\dfrac{2}{3}x+\dfrac{17}{3}.....(i)$
On comparing equation $(i)$ with $y=mx+c$ , we get $m=\dfrac{2}{3}$ . Hence, the slope of the line $2x-3y+17=0$ is equal to $\dfrac{2}{3}$ .
Now, we will find the slope of the line passing through the points $\left( 7,17 \right)$ and $\left( 15,\beta \right)$. Let the slope be ${{m}_{\bot }}$ . We know, the slope of the line joining two points \[({{x}_{1}},{{y}_{1}})\]and \[({{x}_{2}},{{y}_{2}})\] is given as \[m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]. So, the slope of the line passing through the points $\left( 7,17 \right)$ and $\left( 15,\beta \right)$ is given as ${{m}_{\bot }}=\dfrac{\beta -17}{15-7}$ .
$\Rightarrow {{m}_{\bot }}=\dfrac{\beta -17}{8}$
Now, we know that the product of slopes of two lines that are perpendicular to each other is equal to $-1$ . So, $m\times {{m}_{\bot }}=-1$ .
$\Rightarrow \dfrac{2}{3}\times \dfrac{\beta -17}{8}=-1$
$\Rightarrow \dfrac{\beta -17}{8}=-\dfrac{3}{2}$
$\Rightarrow \dfrac{17-\beta }{8}=\dfrac{3}{2}$
On cross-multiplication, we get $2\left( 17-\beta \right)=24$ .
$\Rightarrow 34-2\beta =24$
$\Rightarrow 2\beta =10$
$\Rightarrow \beta =5$
Hence, the value of $\beta $ such that the line passing through the points $\left( 7,17 \right)$ and $\left( 15,\beta \right)$ is perpendicular to the line $2x-3y+17=0$ is given as $\beta =5$.
Hence, the correct option is option D.

Note: While simplifying the equations , please make sure that sign mistakes do not occur. These mistakes are very common and can cause confusions while solving. Ultimately the answer becomes wrong. So , sign conventions should be carefully taken .