Circles
17.0 Equation of chord of circle whose midpoint is given
17.0 Equation of chord of circle whose midpoint is given
When midpoint $M({x_1},{y_1})$ of chord $PQ$ is given, the equation of chord can be find out using $T = {S_1}$ i.e., $S \equiv {x^2} + {y^2} + 2gx + 2fy + c = 0$ and ${S_1} \equiv {x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c = 0$.
Therefore, equation of chord of circle is $$x{x_1} + y{y_1} + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) + c = {x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1} + c$$
or, $$x{x_1} + y{y_1} + g\left( {x + {x_1}} \right) + f\left( {y + {y_1}} \right) = {x_1}^2 + {y_1}^2 + 2g{x_1} + 2f{y_1}$$
NOTE: $T = {S_1}$ is applicable for parabola, ellipse and hyperbola to find equation of chord whose midpoint is given.
Question 26. Find the locus of mid-point of the chord of the circle ${x^2} + {y^2} = 25$ such that it always passes through a fix point $(1,1)$.
Solution: Let us assume the mid-point of chord $PQ$ be $M(h,k)$. The equation of chord can be find out using $T = {S_1}$ i.e.,
$$hx + ky - 25 = {h^2} + {k^2} - 25$$
Since the equation always passes through a fix point $(1,1)$, we get $$h \times 1 + k \times 1 = {h^2} + {k^2}$$ $$h + k = {h^2} + {k^2}$$
Therefore, the locus of mid-point of the chord is a circle ${x^2} + {y^2} - x - y = 0$.
