Question

If the line \(y=mx+c\) makes intercepts \(a,b\) on coordinate axes respectively, such that \(a+b=10\) and \(ab=24\) with \(a>b\), then \(3m+5c=\)

• A. 26
• B. 22
• C. 20
• D. 18

Hint:

For intercept problems, first determine the intercepts and then convert to slope-intercept form.

Answer 1

The Correct Option is D

Concept: The intercept form of a line is \[ \frac{x}{a}+\frac{y}{b}=1. \] Given \[ a+b=10,\qquad ab=24. \]

Step 1: Find the intercepts. \[ t^2-10t+24=0 \] \[ (t-6)(t-4)=0 \] Since \(a>b\), \[ a=6,\qquad b=4. \]

Step 2: Find equation of line. \[ \frac{x}{6}+\frac{y}{4}=1 \] \[ 2x+3y=12 \] \[ y=-\frac23x+4 \] Hence \[ m=-\frac23,\qquad c=4. \]

Step 3: Compute \(3m+5c\). \[ 3m+5c \] \[ =3\left(-\frac23\right)+5(4) \] \[ =-2+20 \] \[ =18 \]