Question

The straight line 2x-3y+17=0 is perpendicular to the line passing through the points
$\left( 7,\text{ }17 \right)\text{ and (15, }\beta \text{) then find }\beta \text{?}$
(a)-5
$(b)-\dfrac{35}{3}$
$(c)\dfrac{35}{3}$
(d) 5

Answer 1

Hint: First find the slopes of lines using the 2 formulas given below and then name the slopes as a and b. Use the conditions if two straight lines with slope a, b are perpendicular then
$a\times b=-1$.
Use this for the first line. If the slope of a line with equation: ax + by + c = 0 is m, then:
$m=-\dfrac{a}{b}$
Use this for the second line. If the slope of a line passing through the points (x, y) and (a, b) is m, then:
$m=\dfrac{b-y}{a-x}$

Complete step-by-step answer:
So firstly, we need to find slopes of both the lines.
Let us assume:

& AB\equiv 2x-3y+17=0 \\\ & CD\equiv \text{line passing through the point }\left( 7,17 \right)\text{ and }\left( 15,\beta \right) \\\ \end{aligned}$$ First we will find the slope of line AB. Use the condition: If the slope of a line with equation: ax + by + c = 0 is m, then: $$m=-\dfrac{a}{b}$$ Let the slope of AB be p. So applying above condition to AB ( 2x - 3y + 17 = 0), From AB we can say a=2 and b=-3. Substituting values of a, b in the slope condition, we get: $$p=-\dfrac{2}{-3}=\dfrac{2}{3}.....\left( 1 \right)$$ If the slope of a line passing through the points (x, y) and (a, b) is m, then: $$m=\dfrac{b-y}{a-x}.....\left( 2 \right)$$ By using the above condition we can find the slope of CD. Let the slope be q. By looking at CD we can say that: x = 7, y = 17, a = 15, $$b=\beta $$, By substituting values of x, y, a, b, q into the condition (2), we get: $$q=\dfrac{b-y}{a-x}=\dfrac{\beta -17}{15-7}=\dfrac{\beta -17}{8}.....\left( 3 \right)$$ We know by condition of perpendicular lines: If two straight lines with slope a, b are perpendicular then $$a\times b=-1$$ By applying above condition, we get: $$p\times q=-1.....\left( 4 \right)$$ By substituting equation (3) equation (2) into equation (4), we get: $$\dfrac{2}{3}\times \dfrac{\beta -17}{8}=-1$$ By cross multiplying, we get: $$\begin{aligned} & 2\left( \beta -17 \right)=-24 \\\ & 2\beta -34=-24 \\\ \end{aligned}$$ By simplifying, we get: $$\begin{aligned} & 2\beta =10 \\\ & \beta = 5 \\\ \end{aligned}$$ $$\therefore $$Option (d) is correct. Note: Be careful with negative signs, as the slope formula of the first line already has a negative sign if you confuse it with the wrong answer.