Home
Class 12
MATHS
If A O Ba n dC O D are two straight line...

If `A O Ba n dC O D` are two straight lines which bisect one another at right angles, show that the locus of a points `P` which moves so that `P AxP B=P CxP D` is a hyperbola. Find its eccentricity.

Text Solution

Verified by Experts

The correct Answer is:
`e=sqrt2`
Taking AOB and COD as the x- and y-axes and their point of intersection O as the origin, clearly, O is the midpoint of AB and CD. Let A be (a, 0) and C be (0, c). Then B is `(-a, 0)` and D is `(0, -c)`.

Let `P-=(x,y)`. Given that
`PA*PB=PC*PD`
`"or "sqrt((x-a)^(2)+y^(2))sqrt((x+a)^(2)+y^(2))=sqrt(x^(2)+(y-c)^(2))sqrt(x^(2)+(y+c)^(2))`
After simplification, we have
`x^(2)-y^(2)=((a^(2)-c^(2)))/(2)`
which is a rectangular hyperbola with eccentricity `sqrt2`.
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.3|10 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.4|5 Videos

  • HYPERBOLA

    CENGAGE|Exercise Exercise 7.1|3 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 straight line segment of length/moves with its ends on two mutually perpendicular lines. Find the locus of the point which divides the line segment in the ratio 1:2

Find the locus of the point at which two given portions of the straight line subtend equal angle.

The coordinates of the point Aa n dB are (a,0) and (-a ,0), respectively. If a point P moves so that P A^2-P B^2=2k^2, when k is constant, then find the equation to the locus of the point Pdot

A straight line L on the xy-plane bisects the angle between O Xa n dO Ydot What are the direction cosines of L ? a. > b. > c. > d. >

Through the vertex 'O' of parabola y^2=4x , chords OP and OQ are drawn at right angles to one another. Show that for all positions of P, PQ cuts the axis of the parabola at a fixed point. Also find the locus of the middle point of PQ.

Find the locus of a point P which moves such that its distance from the line y=sqrt(3)x-7 is the same as its distance from (2sqrt(3),-1)

Through point P(-1,4) , two perpendicular lines are drawn which intersect x-axis at Q and R. find the locus of incentre of Delta PQR .

If P N is the perpendicular from a point on a rectangular hyperbola x y=c^2 to its asymptotes, then find the locus of the midpoint of P N

A line of fixed length a+b moves so that its ends are always on two fixed perpendicular straight lines. Then the locus of the point which divides this line into portions of length aa n db is a/an (a)ellipse (b) parabola (c)straight line (d) none of these