Question

A straight line \(ax+by=1\) passes through the points \((1,2)\) and \((2,1)\). Then \(9(a^{2}+b^{2})=\)

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

If two points lie on a line, substitute both points to obtain simultaneous equations.

Answer 1

The Correct Option is B

Concept: Since both given points lie on the line \(ax+by=1\), substitute them to obtain equations in \(a\) and \(b\).

Step 1: Form equations. Using \((1,2)\), \[ a+2b=1 \] Using \((2,1)\), \[ 2a+b=1 \]

Step 2: Solve for \(a\) and \(b\). Subtracting, \[ a-b=0 \] \[ a=b \] Substituting into \(a+2b=1\), \[ 3a=1 \] \[ a=b=\frac13 \]

Step 3: Calculate \(9(a^2+b^2)\). \[ 9\left(\frac19+\frac19\right) \] \[ =9\cdot\frac29 \] \[ =2 \]