The radius of the circle passing through the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the centre of the circle be $O = (x, y)$. Since the circle passes through $A(-1, 1)$,$B(2, -1)$,and $C(1, 0)$,we have $OA^2 = OB^2 = OC^2 = r^

Vedclass · Accepted Answer

$\frac{\sqrt{130}}{2}$

Vedclass · Answer

(B) Let the centre of the circle be $O = (x, y)$. Since the circle passes through $A(-1, 1)$,$B(2, -1)$,and $C(1, 0)$,we have $OA^2 = OB^2 = OC^2 = r^2$.From $OA^2 = OB^2$:$(x + 1)^2 + (y - 1)^2 = (x - 2)^2 + (y + 1)^2$$x^2 + 2x + 1 + y^2 - 2y + 1 = x^2 - 4x + 4 + y^2 + 2y + 1$$6x - 4y = 3$ ... $(i)$From $OA^2 = OC^2$:$(x + 1)^2 + (y - 1)^2 = (x - 1)^2 + y^2$$x^2 + 2x + 1 + y^2 - 2y + 1 = x^2 - 2x + 1 + y^2$$4x - 2y = -1$ ... $(ii)$Multiplying $(ii)$ by $2$,we get $8x - 4y = -2$ ... $(iii)$Subtracting $(i)$ from $(iii)$:$2x = -5 \implies x = -\frac{5}{2}$Substituting $x$ in $(ii)$:$4(-\frac{5}{2}) - 2y = -1 \implies -10 - 2y = -1 \implies 2y = -9 \implies y = -\frac{9}{2}$Centre $O = (-\frac{5}{2}, -\frac{9}{2})$.Radius $r = \sqrt{(x - 1)^2 + (y - 0)^2} = \sqrt{(-\frac{5}{2} - 1)^2 + (-\frac{9}{2})^2} = \sqrt{(-\frac{7}{2})^2 + (-\frac{9}{2})^2} = \sqrt{\frac{49}{4} + \frac{81}{4}} = \sqrt{\frac{130}{4}} = \frac{\sqrt{130}}{2}$.

The radius of the circle passing through the points $(-1, 1)$,$(2, -1)$,and $(1, 0)$ is

Similar Questions

The circle $4x^2+4y^2-12x-12y+9=0$

If the arcs of the same length in two circles $S_1$ and $S_2$ subtend angles $75^{\circ}$ and $120^{\circ}$ respectively at the center,then the ratio of their radii $\frac{r_1}{r_2}$ is equal to:

$A$ triangle $PQR$ is inscribed in the circle $x^2 + y^2 = 25$. If the coordinates of $Q$ and $R$ are $(3, 4)$ and $(-4, 3)$ respectively,then $\angle QPR = \dots$

Let $PQ$ and $RS$ be the tangents at the extremities of the diameter $PR$ of a circle of radius $r$. If $PS$ and $RQ$ intersect at a point $X$ on the circumference of the circle,then $(PQ \cdot RS)$ is equal to

For the circle $x^2 + y^2 + 6x - 8y + 9 = 0$,which of the following statements is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module