Home
Class 12
MATHS
If P(x1,y1),Q(x2,y2),R(x3,y3) and S(x4...

If `P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3) and S(x_4,y_4)` are four concyclic points on the rectangular hyperbola and `xy = c^2` , then coordinates of the orthocentre ofthe triangle `PQR` is

Text Solution

Verified by Experts

We know that the orthocentre of the triangle formed by points `P(t_(1)), Q(t_(2)) and R(t_(3))` on hyperbola lies on the same hyperbola and has coordinates `((-c)/(t_(1)t_(2)t_(3)),-ct_(1)t_(2)t_(3))`.
Also, if points `P(t_(1)), Q(t_(2)),R(t_(3)) and S(t_(4))` are concyclic then `t_(1)t_(2)t_(3)t_(4)=1.`
Therefore, orhtocentre of triangle PQR is `(-ct_(4),(-c)/(t_(4)))-=(-x_(4),-y_(4)).`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE|Exercise Exercise (Single)|68 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise (Multiple)|18 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.5|5 Videos

  • HIGHT AND DISTANCE

    CENGAGE|Exercise JEE Previous Year|3 Videos

  • INDEFINITE INTEGRATION

    CENGAGE|Exercise Question Bank|21 Videos

Similar Questions

Explore conceptually related problems

A(6,1), B(8,2) and C(9,4) are three vertices of a parallelograam ABCd taken in order. Find the fourth vertex D. IF (x_(1),y_(1)), (x_(2),y_(2)), (x_(3),y_(3)) and (x_(4),y_(4)) are the four vertices of the parallelgraam then using the given points, find the value of (x_(1)+x_(3)-x_(2),y_(1)+y_(3)+y_(2)) and state the reason for your result.

The equation of the chord joining two points (x_(1),y_(1)) and (x_(2),y_(2)) on the rectangular hyperbola xy=c^(2) , is

Let P(x_1, y_1) and Q(x_2, y_2), y_1 < 0, y_2 < 0 , be the end points of the latus rectum of the ellipse x^2+4y^2 = 4 . The equations of parabolas with latus rectum PQ are

Let P(x_1, y_1) and Q(x_2, y_2), y_1 < 0, y_2 < 0 , be the end points of the latus rectum of the ellipse x^2+4y^2 = 4 . The equations of parabolas with latus rectum PQ are

If points Aa n dB are (1, 0) and (0, 1), respectively, and point C is on the circle x^2+y^2=1 , then the locus of the orthocentre of triangle A B C is (a) x^2+y^2=4 (b) x^2+y^2-x-y=0 (c) x^2+y^2-2x-2y+1=0 (d) x^2+y^2+2x-2y+1=0

If sum_(i-1)^4(xi2+y i2)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 the vertices of a rectangle collinear the vertices of a trapezium none of these

Consider the parabola y^(2)=4x , let P and Q be two points (4,-4) and (9,6) on the parabola. Let R be a moving point on the arc of the parabola whose x-coordinate is between P and Q. If the maximum area of triangle PQR is K, then (4K)^(1//3) is equal to

If line x-2y-1=0 intersects parabola y^(2)=4x at P and Q, then find the point of intersection of normals at P and Q.

The coordinates of two points Pa n dQ are (x_1,y_1)a n d(x_2,y_2)a n dO is the origin. If the circles are described on O Pa n dO Q as diameters, then the length of their common chord is (a) (|x_1y_2+x_2y_1|)/(P Q) (b) (|x_1y_2-x_2y_1|)/(P Q) (|x_1x_2+y_1y_2|)/(P Q) (d) (|x_1x_2-y_1y_2|)/(P Q)

Consider a hyperbola xy = 4 and a line y = 2x = 4 . O is the centre of hyperbola. Tangent at any point P of hyperbola intersect the coordinate axes at A and B. Locus of circumcentre of triangle OAB is