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 find coordinates of the orthocentre of the triangle `PQR`

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 PUBLICATION|Exercise EXERCISES|68 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWERS TYPE|18 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 7.5|5 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

Find the coordinates of the foci of the rectangular hyperbola x^(2)-y^(2) = 9

Find the coordinates of the foci and the center of the hyperbola, x^2-3y^2-4x=8

If tangents O Q and O R are dawn to variable circles having radius r and the center lying on the rectangular hyperbola x y=1 , then the locus of the circumcenter of triangle O Q R is (O being the origin). (a) x y=4 (b) x y=1/4 x y=1 (d) none of these

The equation to the chord joining two points (x_1,y_1)a n d(x_2,y_2) on the rectangular hyperbola x y=c^2 is:

The slope of the normal to the parabola 3y^(2)+4y+2=x at a point it is 8. find the coordinates of the points.

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.

If P (1, 2), Q (4, 6), R (5, 7) and S (x, y) are the successive four vertices of the parallelogram PQRS, then

the coordinates of the vertices A,B,C of the triangle ABC are (7,-3) , (x,8) and (4,y) respectively , if the coordinates of the centroid of the triangle be (2,-1) , find x and y.

Lines x + y = 1 and 3y = x + 3 intersect the ellipse x^(2) + 9y^(2) = 9 at the points P , Q , R . The area of the triangle PQR is _

Find the equation of the normal to the hyperbola 3x^(2)-4y^(2)=12 at the point (x_(1),y_(1)) on it. Hence, show that the straight line x+y+7=0 is a normal to the hyperbola. Find the coordinates of the foot of the normal.