Question

If a line with slope 3 passing through the point \((4,-5)\) makes intercepts \(a,b\) on the coordinate axes, then \(3a^2+4b^2=\)

• A. 1445
• B. \(\frac{3757}{3}\)
• C. 2465
• D. \(\frac{4139}{3}\)

Hint:

Use point-slope form first and then obtain intercepts directly.

Answer 1

The Correct Option is A

Step 1: Find equation of the line. Given slope \(m=3\) and point \((4,-5)\), \[ y+5=3(x-4) \] \[ y=3x-17 \]

Step 2: Find intercepts. For x-intercept, \[ 0=3x-17 \] \[ a=\frac{17}{3} \] For y-intercept, \[ b=-17 \]

Step 3: Calculate \(3a^2+4b^2\). \[ 3\left(\frac{17}{3}\right)^2+4(17)^2 \] \[ =3\cdot\frac{289}{9}+1156 \] \[ =\frac{289}{3}+1156 \] \[ =\frac{3757}{3} \]