Question

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

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 $$(x,y)$$.
The coordinates of A$$(x_1,y_1)$$and B$$(x_2,y_2)$$ 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 $$(x_2,y_2)$$in the ratio m:n internally is given by
P = $$({\displaystyle \frac{m{x}_2+n{x}_1}{m+n}},{\displaystyle \frac{m{y}_2+n{y}_1}{m+n}})$$
Substituting the values of (m, n) and coordinates of A and B in the above equation gives,
$\Rightarrow$$$({\displaystyle \frac{1(2)+(-2)(-1)}{1-2}},{\displaystyle \frac{1(0)+(-2)(0)}{1-2}})$$
$$\Rightarrow$$ $$(-4,0)$$
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).