Question

The sum of the slopes of two lines passing through the origin is zero. If one line passes through $(a,4)$ and another line passes through $(-3,b)$, then which of the following can be the values of a and b?

• A. $a=2,b=6$
• B. $a=4,b=3$
• C. All of the above
• D. $a=-3,b=-4$

Answer 1

The correct option is C All of the above

Detailed step-by-step solution:
Since both the lines pass through the origin, the equation of the lines is in the form of $y=mx$.
Let the equation of the first line be $y=m_{1}x$, and the equation of the second line be $y=m_{2}x$.
Slope of the first line: $y=m_{1}x$
slope $=m_1$ (slope = m in the equation $y=mx$)
Slope of the second line: $y=m_{2}x$
slope $=m_2$ (slope = m in the equation $y=mx$)
(Slope of the first line) + (Slope of the second line) $=0$
$\Rightarrow m_1+m_2=0$
$\Rightarrow m_2=-m_1$
Let $m_1=m.$
$\Rightarrow m_2=-m$
Equation of the first line: $y=mx$
Equation of the second line: $y=-mx$
Line 1 passes through $(a,4)$.
$4=am$ (substituting $x=a$ and $y=4$ in $y=mx$)
$\Rightarrow a=\frac{4}{m}$
$\Rightarrow b=-m(-3)$ (substituting $x=-3$ and $y=b$ in $y=-mx$)
$\Rightarrow b=3m$
Multiplying a and b, we get:
$a\times b=\frac{4}{m}\times \left(3m\right)$
$\Rightarrow ab=12$

Let’s check the options.
A. $a=2,b=6$: Substituting in $ab=12$, we get:
$2\times 6=12$
$2$ and $6$ can be the values of a and b, respectively.
B. $a=4,b=3$: Substituting in $ab=12$, we get:
$4\times 3=12$
$4$ and $3$ can be the values of a and b, respectively.
C. $a=-3,b=-4$: Substituting in $ab=12$, we get:
$(-3)\times (-4)=12$
$-3$ and $-4$ can be the values of a and b, respectively.
Option D is correct.