If the points (a1, b1), (a2, b2) and (a1 + a2, b1 + b2) are collinear, then

Question

If the points (a1, b1), (a2, b2) and (a1 + a2, b1 + b2) are collinear, then (A) a1b2=a2b1 (B) a1+ a2 = b1 + b2 (C) a2b2=a1b1 (D) a1+b1=a2+b2

Tardigrade Admin · Accepted Answer

The given points are collinear. ∴ (1/2)| beginmatrixa1&b1&1 a2&b2&1 a1+a2&b1+b2&1 endmatrix|=0 Applying R2arrow R2- R1, R3arrow R3- R1, we get | beginmatrixa1&b1&1 a2-a1&b2-b1&0 a2&b2&0 endmatrix|=0 Expanding along C3, we get b2(a2-a1)-a2(b2-b1)=0 ⇒ -a1b2+a2b1=0 ⇒ a1b2=a2b1

Q. If the points $(a_{1}, b_{1})$ , $(a_{2}, b_{2})$ and $(a_{1} + a_{2}, b_{1} + b_{2})$ are collinear, then

$a_{1} b_{2} = a_{2} b_{1}$

$a_{1} + a_{2} = b_{1} + b_{2}$

$a_{2} b_{2} = a_{1} b_{1}$

$a_{1} + b_{1} = a_{2} + b_{2}$

Q. If the points (a1​,b1​), (a2​,b2​) and (a1​+a2​,b1​+b2​) are collinear, then

a1​b2​=a2​b1​

a1​+a2​=b1​+b2​

a2​b2​=a1​b1​

a1​+b1​=a2​+b2​

Q. If the points $(a_{1}, b_{1})$ , $(a_{2}, b_{2})$ and $(a_{1} + a_{2}, b_{1} + b_{2})$ are collinear, then

$a_{1} b_{2} = a_{2} b_{1}$

$a_{1} + a_{2} = b_{1} + b_{2}$

$a_{2} b_{2} = a_{1} b_{1}$

$a_{1} + b_{1} = a_{2} + b_{2}$