Home
Class 12
MATHS
If the circle x^2+y^2=a^2 intersects the...

If the circle `x^2+y^2=a^2` intersects the hyperbola `x y=c^2` at four points `P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3),` and `S(x_4, y_4),` then

A

`x_(1)+x_(2)+x_(3)+x_(4)=0`

B

`y_(1)+y_(2)+y_(3)+y_(4)=0`

C

`x_(1)x_(2)x_(3)x_(4)=c^(4)`

D

`y_(1)y_(2)y_(3)y_(4)=c^(4)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
Solving `xy=c^(2) and x^(2)+y^(2)=a^(2)`, we have
`x^(2)+(c^(4))/(x^(2))=a^(2)`
`"or "x^(4)-a^(2)x^(2)+c^(4)=0`
`therefore" "Sigmax_(i)=0 and x_(1)x_(2)x_(3)x_(4)=c^(4)`
Similarly, if we eliminate y, then `Sy_(i)=0 and y_(1)y_(2)y_(3)y_(4)=c^(4)` .
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise COMOREHENSION TYPE|21 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise MATRIX MATHC TYPE|10 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise EXERCISES|68 Videos

  • HIGHT AND DISTANCE

    CENGAGE PUBLICATION|Exercise Archives|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE PUBLICATION|Exercise Multiple Correct Answer Type|2 Videos

Similar Questions

Explore conceptually related problems

If the normals to the ellipse x^2/a^2+y^2/b^2= 1 at the points (x_1, y_1), (x_2, y_2) and (x_3, y_3) are concurrent, prove that |(x_1,y_1,x_1y_1),(x_2,y_2,x_2y_2),(x_3,y_3,x_3y_3)|=0 .

Suppose the circle having equation x^2+y^2=3 intersects the rectangular hyperbola x y=1 at points A ,B ,C ,a n dDdot The equation x^2+y^2-3+lambda(x y-1)=0,lambda in R , represents.

Knowledge Check

  • If the points of intersection of the parabola y^(2) = 4ax and the circle x^(2) + y^(2) + 2gx + 2fy + c = 0 are (x_(1), y_(1)) ,(x_(2) ,y_(2)) ,(x_(3), y_(3)) and (x_(4), y_(4)) respectively , then _

    A
    `y _(1) + y_(2) + y_(3)+y_(4) = 0 `
    B
    `sqrt(x)_(1)+sqrt(x)_(2)+sqrt(x)_(3)+sqrt(x)_(4) = 0 `
    C
    `y _(1) - y_(2) + y_(3) - y _(4) = 0 `
    D
    `y _(1) - y_(2) - y_(3) + y _(4) = 0 `

  • The circle x^2+y^2+x+y=0 and x^2+y^2+x-y=0 intersect at an angle of :

    A
    `pi/6`
    B
    `pi/4`
    C
    `pi/3`
    D
    `pi/2`

  • If x_1, x_2 , x_3 and y_1 , y_2 , y_3 are both in G.P. with the same common ratio, then the points (x_1 , y_1),(x_2 , y_2) and (x_3 , y_3)

    A
    are vertices of a triangle
    B
    lie on a straight line
    C
    lie on an ellipse
    D
    lie on a circle

    • Similar Questions

    Explore conceptually related problems

    If sum_(i=1)^4(x1^2+y 1^2)lt=2x_1x_3+2x_2x_4+2y_2y_3+2y_1y_4, the points (x_1, y_1),(x_2,y_2),(x_3, y_3),(x_4,y_4) are

    If the tangent at point P(h, k) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 cuts the circle x^(2)+y^(2)=a^(2) at points Q(x_(1),y_(1)) and R(x_(2),y_(2)) , then the vlaue of (1)/(y_(1))+(1)/(y_(2)) is

    If x_(1),x_(2),x_(3) as well as y_(1),y_(2),y_(3) are in G.P. with the same common ratio , then show that the points (x_(1),y_(1)),(x_(2),y_(2))and(x_(3),y_(3)) lie on a straight line .

    If two circles (x-1)^2+(y-3)^2=r^2 and x^2+y^2-8x+2y+8=0 intersect in two distinct points then

    The slope of the tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point ( x_(1),y_(1)) is-

    Home
    Profile