If one end of a diameter of a circle x²+y²-4x-6y+11=0; is (3,4). Find the coordinate of the other end of the diameter.

If one end of a diameter of a circle x²+y²-4x-6y+11=0; is (3,4). Find the coordinate of the other end of the diameter. 

Solution:

Given,

Equation of circle is x² +y² -4x -6y +11 = 0

One end of the diameter is (3,4)

Let, (3,4)= (x₂,y₂)

To find: Other end of the diameter (x₁,y₁) = ?

We know,

Equation of a circle having two ends of diameter is (x -x₁)(x -x₂) + (y -y₁)(y -y₂) = 0 ---(i)

Solving given equation of circle:

x² +y² -4x -6y +11 = 0

or, x² +y² -4x -6y +3 +8 = 0

or, x² -4x +3 +y² -6y +8 = 0

or, x² -(3+1)x +3 +y² -(4+2)y +8 = 0

or, x² -3x -x +3 +y² -4y -2y +8 = 0

or, x(x-3) -1(x-3) + y(y-4) -2(y-4) = 0

or, (x-1)(x-3) + (y-2)(y-4) = 0 ---(ii)

Comparing equation (ii) with equation (i), we get:

x₁ = 1

x₂ = 3

y₁ = 2

y₂ = 4

So, the coordinates of the diameter on the other end is (x₁,y₁) = (1,2).

Explanation:

As we know the equation of the circle when the coordinates of its diameter are given, we can compare the given equation with that equation. 

To solve the given equation, we first divided 11 into two such parts from which we could solve the front as well as back part of the equation. By front part, I mean x²-4x and by back part, I mean y²-6x. 

To solve it, we have used mid-term factorization method, in which we need to find two such numbers which when multiplied result the constant term (here for front part, constant term is 3 and for back part, constant term is 8). Since, both constant terms 3 and 8 are positive, we also need to get the 'bx' when adding those two numbers.

Here for front part, numbers would be 3 and 1

And, for back part, numbers would be 4 and 2.

This means, 3*1=3, and 3+1=4

Also, 4*2=8 and 4+2 = 6

Now, write the entire equation in factor form as we did above, then compare the equation you receive with the equation of circle. Then, you will receive your desired answer.

Some solutions of coordinate geometry:
The equation of the diagonal of a square is 3x-4y+5=0 and one of its vertices is (1,2). Find the equation of the side of the square that meet at this point.

 

Related Notes and Solutions:

Here is the website link to all the important formulae of Coordinate Geometry of Class 10.

CLASS 10 COORDINATE GEOMETRY QUESTION AND ANSWER • SOLUTION OF QUESTION FROM EQUATION OF CIRCLES IN COORDINATE FORM OF CLASS 10 UNIT COORDINATE GEOMETRY • COORDINATES OF DIAMETER OF A CIRCLE

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