Q. 135.0( 2 Votes )

# If a circle passes through the point (0, 0), (a, 0), (0, b), then find the coordinates of its centre.

Answer :

Given that the circle passes through the points O(0,0), A(a,0) and B(0,b).

Let us first find the length of the sides of the triangle formed by the points OAB.

We know that distance between two points (x_{1},y_{1}) and (x_{2},y_{2}) is .

⇒

⇒

⇒ OA = |a| .....(1)

⇒

⇒

⇒ OB = |b| .....(2)

⇒

⇒ ..... (3)

Now consider OA^{2} + OB^{2},

⇒ OA^{2} + OB^{2} = (|a|)^{2} + (|b|)^{2}

⇒ OA^{2} + OB^{2} = a^{2} + b^{2}

⇒

⇒ OA^{2} + OB^{2} = AB^{2}

We got ABC is a right angled triangle with AB as the hypotenuse.

We know that the circumcentre of a right-angled triangle is the midpoint of the hypotenuse.

⇒ Centre =

⇒ Centre =

∴ The coordinates of the centre is .

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A parabolic reflector is 5 cm deep and its diameter is 20 cm. How far is its focus from the vertex?

RS Aggarwal - Mathematics