Home
Class 12
MATHS
If the chord xcosalpha+ysinalpha=p of th...

If the chord `xcosalpha+ysinalpha=p` of the hyperbola `(x^2)/(16)-(y^2)/(18)=1` subtends a right angle at the center, and the diameter of the circle, concentric with the hyperbola, to which the given chord is a tangent is `d ,` then the value of `d/4` is__________

Text Solution

Verified by Experts

The correct Answer is:
24
The equation of hyperbola is
`(x^(2))/(16)-(y^(2))/(18)=1`
`"or "9x^(2)-8y^(2)-144=0`
Homogenizing this equation using
`(x cosalpha+y sin alpha)/(p)=1`
We have `9x^(2)-8y^(2)-144((x cos alpha+y sin alpha)/(p))^(2)=0`
Since these lines are perpendicular to each other, we have
`9p^(2)-8p^(2)-144(cos^(2)alpha+sin^(2)alpha)=0`
`p^(2)=144or p = pm12`
`therefore" Radius of circle"=12`
`therefore" Diameter of circle"=24`
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise JEE MAIN|3 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise JEE ADVANCED|6 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise MATRIX MATHC TYPE|10 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

The condition that the chord xcos alpha +ysin alpha -p=0 of x^2+y^2-a^2=0 may subtend a right angle at the centre of the circle is

A normal chord of the parabola y^2=4ax subtends a right angle at the vertex, find the slope of chord.

A variable chord of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,(b > a), subtends a right angle at the center of the hyperbola if this chord touches. a fixed circle concentric with the hyperbola a fixed ellipse concentric with the hyperbola a fixed hyperbola concentric with the hyperbola a fixed parabola having vertex at (0, 0).

If the chords of contact of tangents from two points (-4,2) and (2,1) to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are at right angle, then find then find the eccentricity of the hyperbola.

If the chord u=mx +1 of the circel x ^(2) +y^(2)=1 subtends an angle of 45^(@) at the major segment of the circle, then the value of m is-

Let P(4,3) be a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 . If the normal at P intersects the x-axis at (16,0), then the eccentriclty of the hyperbola is

With one focus of the hyperbola x^2/9-y^2/16=1 as the centre, a circle is drawn which is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is

If P Q is a double ordinate of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 such that O P Q is an equilateral triangle, O being the center of the hyperbola, then find the range of the eccentricity e of the hyperbola.

The locus of the midpoints of chords of the circle x^(2) + y^(2) = 1 which subtends a right angle at the origin is

A chord AB of the circle x^(2)+y^(2)=a^(2) subtends a right angle at its centre. Show that the locus of the centroid of the angle PAB as P moves on the circle is another circle.