Question

Harmonic conjugate of (0.0) wrt to (-1,0) and (2,0) is
(1) $\left( 4,0 \right)$
(2) $\left( -4,0 \right)$
(3) $\left( 3,0 \right)$
(4) $\left( -3,0 \right)$

Answer 1

Hint: Applying the harmonic conjugate concept and finding the solution by using point of division method.

Complete step-by-step answer:

We have to find the harmonic conjugate of P(0,0) wrt to A(-1,0) and B(2,0).
Considering the coordinates of P as \[\left( x,y \right)\].
The coordinates of A\[\left( {{x}_{1}},{{y}_{1}} \right)\]and B\[\left( {{x}_{2}},{{y}_{2}} \right)\] are considered as follows.
Harmonic conjugate: The points PQ which divides the line segment AB in the same ratio m : n internally and externally then P and Q are said to be harmonic conjugates of each other wrt A and B.
Formula: If point P divides AB in the ratio m : n internally then harmonic conjugate divides AB in the ratio (-m : n).
P divides AB in the ratio is given by
\[{{x}_{1}}-x\]: \[x-{{x}_{2}}\]
By substituting the values
\[\Rightarrow \]-1 – 0 : 0 – 2
\[\Rightarrow \]-1 : -2
\[\Rightarrow \]Harmonic conjugate divides AB in the ratio is given as
\[\Rightarrow \](-(-1) : -2)
\[\Rightarrow \]1 : -2
So harmonic conjugate divides AB internally in the ratio is (1, -2)
To find the harmonic conjugate point, we have to use the point of division formula.
Formula: The point P which divides the line segment joining the points A \[\left( {{x}_{1,}}{{y}_{1}} \right)\], B \[\left( {{x}_{2}},{{y}_{2}} \right)\]in the ratio m:n internally is given by
P = \[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\]
Substituting the values of (m, n) and coordinates of A and B in the above equation gives,
$\Rightarrow $\[\left( \dfrac{1(2)+(-2)(-1)}{1-2},\dfrac{1(0)+(-2)(0)}{1-2} \right)\]
\[\Rightarrow \] \[\left( -4,0 \right)\]
Therefore the option is (2)

Note: Harmonic conjugate divides the line segment AB externally in the ratio (m : n) or internally in the ratio (-m : n).