Question

in the xy-plane, lines k and j are perpendicular and intersect at the point (4, 23). line j passes through the origin, and line k passes through the point (p, 0), where p is a constant. what is the value of p?
a. -128.25
b. -127.75
c. 135.75
d. 136.25

Explanation:

Step1: Find slope of line j

Line j passes through origin \((0,0)\) and \((4, 23)\). Slope \(m_j=\frac{23 - 0}{4 - 0}=\frac{23}{4}\).

Step2: Find slope of line k

Lines k and j are perpendicular, so \(m_k\times m_j=- 1\). Thus, \(m_k=-\frac{4}{23}\).

Step3: Equation of line k

Line k passes through \((4,23)\) with slope \(-\frac{4}{23}\). Using point - slope form \(y - y_1=m(x - x_1)\), we get \(y - 23=-\frac{4}{23}(x - 4)\).

Step4: Find p when \(y = 0\)

Substitute \(y = 0\) into the equation of line k:
\[

$$\begin{align*} 0-23&=-\frac{4}{23}(p - 4)\\ -23\times(-\frac{23}{4})&=p - 4\\ \frac{529}{4}&=p - 4\\ p&=\frac{529}{4}+4\\ p&=\frac{529 + 16}{4}\\ p&=\frac{545}{4}=136.25 \end{align*}$$

\]

Answer:

d. 136.25