Home
Class 12
MATHS
If the abscissa and ordinates of two poi...

If the abscissa and ordinates of two points `Pa n dQ` are the roots of the equations `x^2+2a x-b^2=0` and `x^2+2p x-q^2=0` , respectively, then find the equation of the circle with `P Q` as diameter.

Text Solution

Verified by Experts

Let `x_(1),x_(2)` and `y_(1),y_(2)` be the roots of `x^(2)+2ax-b^(2)=0` and `x^(2)+2px-q^(2)=0`, respectively. Then,
`x_(1)+x_(2)=-2a,x_(1)x_(2)=-b^(2)`
and `y_(1)+y_(2)=-2p,y_(1)y_(2)=-q^(2)`
The equation of the circle with `P(x_(1),y_(1))` and `Q(x_(2),y_(2))` as the endpoints of diameter is
`(x-x_(1))(x-x_(2))+(y-y_(1))(y-y_(2))=0`
or `x^(2)+y^(2)-x(x_(1)+x_(2))-y(y_(1)+y_(2))+x_(1)x_(2)+y_(1)y_(2)=0`
or `x^(2)+y^(2)+2ax+2py-b^(2)-q^(2)=0`
Promotional Banner

Topper's Solved these Questions

  • CIRCLE

    CENGAGE ENGLISH|Exercise Examples|13 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise CONCEPT APPLICATION EXERCISE 4.1|1 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Matrix|4 Videos

  • CIRCLES

    CENGAGE ENGLISH|Exercise Comprehension Type|8 Videos

Similar Questions

Explore conceptually related problems

If the abscissa and ordinates of two points P and Q are the roots of the equations x^2+2a x-b^2=0 and x^2+2p x-q^2=0 , respectively, then find the equation of the circle with P Q as diameter.

Find the roots of the equations. Q. x^(2)-2x+5=0

Find the roots of the equations. Q. 2x^(2)+x-3=0

Find the roots of the equations. Q. 5x^(2)-x+4=0 .

The abscissa of the two points A and B are the roots of the equation x^2+2a x-b^2=0 and their ordinates are the roots of the equation x^2+2p x-q^2=0. Find the equation of the circle with AB as diameter. Also, find its radius.

The abscissa of the two points A and B are the roots of the equation x^2+2a x-b^2=0 and their ordinates are the roots of the equation x^2+2p x-q^2=0. Find the equation of the circle with AB as diameter. Also, find its radius.

The abscissae of two points A and B are the roots of the equaiton x^2 + 2x-a^2 =0 and the ordinats are the roots of the equaiton y^2 + 4y-b^2 =0 . Find the equation of the circle with AB as its diameter. Also find the coordinates of the centre and the length of the radius of the circle.

If the abscissae of points A, B are the roots of the equation, x^(2) + 2ax - b^(2) = 0 and ordinates of A, B are roots of y^(2)+2py- q^(2) = 0 , then find the equation of a circle for which overline(AB) is a diameter.

If the roots of the quadratic equation x^2+p x+q=0 are tan30^0 and tan15^0, respectively, then find the value of q-p .

If the roots of the quadratic equation x^2+p x+q=0 are tan30^0a n dtan15^0, respectively, then find the value of 2+q-pdot