Answer

Verified

438k+ views

**Hint:**We will first let point be $\left( {h,k} \right)$ whose distance from the point $\left( {1,0} \right)$ to the distance from the point $\left( { - 1,0} \right)$ is equal to $\dfrac{1}{3}$. Find the distance of both the given points from $\left( {h,k} \right)$ and use the given ratio to form the equation of circumcircle which passes through the vertices of the triangle ABC. Compare the equation with the standard equation of the circle to find the circumcentre.

**Complete step-by-step answer:**Let the coordinates of the point be $\left( {h,k} \right)$ whose distance from the point $\left( {1,0} \right)$ to the distance from the point $\left( { - 1,0} \right)$ is equal to $\dfrac{1}{3}$.

We will first find the distance from $\left( {h,k} \right)$ to $\left( {1,0} \right)$

If $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are two points, then the distance between the points is given by $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $

Then, the distance from $\left( {h,k} \right)$ to $\left( {1,0} \right)$ will be $\sqrt {{{\left( {h - 1} \right)}^2} + {k^2}} $

Similarly, the distance from $\left( {h,k} \right)$ to $\left( { - 1,0} \right)$ will be $\sqrt {{{\left( {h + 1} \right)}^2} + {k^2}} $

We are given that the ratio of both the distances is $\dfrac{1}{3}$

Hence, $\dfrac{{\sqrt {{{\left( {h - 1} \right)}^2} + {k^2}} }}{{\sqrt {{{\left( {h + 1} \right)}^2} + {k^2}} }} = \dfrac{1}{3}$

On squaring both sides, we will get,

$\dfrac{{{{\left( {h - 1} \right)}^2} + {k^2}}}{{{{\left( {h - 1} \right)}^2} + {k^2}}} = \dfrac{1}{9}$

Cross-multiply and simplify

$

9\left( {{{\left( {h - 1} \right)}^2} + {k^2}} \right) = {\left( {h - 1} \right)^2} + {k^2} \\

\Rightarrow 9\left( {{h^2} + 1 - 2h + {k^2}} \right) = {h^2} + 1 + 2h + {k^2} \\

$

On solving the brackets, we get

$

9\left( {{h^2} + 1 - 2h + {k^2}} \right) = {h^2} + 1 + 2h + {k^2} \\

\Rightarrow 9{h^2} + 9 - 18h + 9{k^2} = {h^2} + 1 + 2h + {k^2} \\

\Rightarrow 8{h^2} + 8{k^2} - 20h + 8 = 0 \\

\Rightarrow {h^2} + {k^2} - \dfrac{5}{2}h + 1 = 0 \\

$

We have to find the circumcentre of the triangle ABC.

Circumference is the centre of the circle which passes through each point on the circle.

The equation ${h^2} + {k^2} - \dfrac{5}{2}h + 1 = 0$ represents the equation of the circle.

Compare it with the standard equation of the circle to find the circumcentre.

The standard equation of the circle is ${x^2} + {y^2} + 2gx + 2fy + c = 0$, whose centre is $\left( { - g, - f} \right)$

Here, $2g = - \dfrac{5}{2}$ and $2f = 0$

Hence, $g = - \dfrac{5}{4}$ and $f = 0$

Therefore, the coordinates of circumcentre is $\left( {\dfrac{5}{4},0} \right)$

**Hence, option A is correct.**

**Note:**Since, we have to find the circumcentre of the triangle with vertices A,B and C, the coordinates of any vertex A,B or C will satisfy the equation of the circumcircle. Also, the general equation of circle is ${x^2} + {y^2} + 2gx + 2fy + c = 0$, where $\left( { - g, - f} \right)$ is the centre of the circle and $\sqrt {{g^2} + {f^2} - c} $ is the radius of the circle.

Recently Updated Pages

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What organs are located on the left side of your body class 11 biology CBSE