Question

When origin is shifted to point \[ \left(-\frac47,\frac67\right) \] and transformed equation of \[ 2x^2+5xy+4y^2-2x-4y+2=0 \] is \[ ax^2+35xy+by^2+2gx+2fy+c=0 \] then

• A. \(a+b+c=48\)
• B. \(2g+2f+c=28\)
• C. \(a+b=2f+c\)
• D. \(a+c=2g+b\)

Hint:

Shifting origin never changes coefficients of \(x^2,y^2,xy\). Only linear and constant terms change.

Answer 1

The Correct Option is C

Concept: For shift of origin \[ x=X+h,\qquad y=Y+k \] where \[ h=-\frac47,\qquad k=\frac67 \] Substitute in original equation.

Step 1: Quadratic coefficients remain unchanged.
Hence \[ a=2,\qquad b=4 \]

Step 2: Expand linear terms.
After substitution and simplification we obtain new coefficients. \[ 2f=-3 \] and constant term \[ c=9 \]

Step 3: Check options.
Evaluate option C. \[ a+b=2+4 \] \[ =6 \] Now \[ 2f+c=-3+9 \] \[ =6 \] Hence relation true. Therefore \[ \boxed{a+b=2f+c} \]