Question

The equation of a line through $(3,-4)$ and perpendicular to the line $3x+4y=5$ is:

• A. $4x+3y=24$
• B. $y-4=(x+3)$
• C. $3y-4x=24$
• D. $(y+4)=\frac{4}{3}(x-3)$

Answer 1

The correct option is D

$(y+4)=\frac{4}{3}(x-3)$

Determine the equation of the line :

Let the line be $AB$.

Given that: The line passes through $(3,4)$ and perpendicular to the line $3x+4y=5$.

The given line is $3x+4y=5$

Slope of $3x+4y=5$ is:

$\begin{array}{rcl}3x+4y& =& 5\\ & \Rightarrow & 4y=-3x+5\\ & \Rightarrow & y=\frac{-3}{4}x+5\\ & & \\ \therefore slope& =& -\frac{3}{4}\end{array}$

Slope of $AB$:

$\begin{array}{rcl}& =& -\frac{1}{{\displaystyle \frac{-3}{4}}}[\because m_1\times m_2=-1]\\ & =& \frac{4}{3}\end{array}$

Equation of $AB$:

$$\begin{gathered} y-y_1=m(x-x_1) \\ \Rightarrow y-(-4)=\frac{4}{3}(x-3) \\ \Rightarrow y+4=\frac{4}{3}(x-3) \end{gathered}$$

Therefore, the correct answer is Option (D)