Home
Class 12
MATHS
A normal to the hyperbola (x^2)/(a^2)-(y...

A normal to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` meets the axes at `Ma n dN` and lines `M P` and `N P` are drawn perpendicular to the axes meeting at `Pdot` Prove that the locus of `P` is the hyperbola `a^2x^2-b^2y^2=(a^2+b^2)dot`

Text Solution

Verified by Experts

The equation of normal at the point `Q(a sec phi, b tan phi)` to the hyperbola
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
is `ax cos phi+by cot phi=a^(2)+b^(2)" (1)"`
The normal (1) meets the x-axis at
`M((a^(2)+b^(2))/(a) sec phi, 0)`
and the y-axis at
`N(0,(a^(2)+b^(2))/(b)tan phi)`
Let point P be (h,k). From the diagram,

`h=(a^(2)+b^(2))/(a)sec phi`
and `k=(a^(2)+b^(2))/(b) tan phi`
Eliminating `phi` by using the relation `sec^(2)phi-tan^(2)phi=1`, we have
`((ah)/(a^(2)+b^(2)))^(2)-((bk)/(a^(2)+b^(2)))^(2)=1`
Therefore, `a^(2)x^(2)-b^(2)y^(2)=(a^(2)+b^(2))^(2)` is the required locus of P.
Promotional Banner

Topper's Solved these Questions

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 7.6|4 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise EXERCISES|68 Videos

  • HYPERBOLA

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 7.4|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

If the normal at P(theta) on the hyperbola (x^2)/(a^2)-(y^2)/(2a^2)=1 meets the transvers axis at G , then prove that A GdotA^(prime)G=a^2(e^4sec^2theta-1) , where Aa n dA ' are the vertices of the hyperbola.

Ifthe normal at P to the rectangular hyperbola x^2-y^2=4 meets the axes in G and g and C is the centre of the hyperbola, then

If the normal at a point P to the hyperbola x^2/a^2 - y^2/b^2 =1 meets the x-axis at G , show that the SG = eSP.S being the focus of the hyperbola.

For hyperbola x^2/a^2-y^2/b^2=1 , let n be the number of points on the plane through which perpendicular tangents are drawn.

A normal to the hyperbola, 4x^2_9y^2 = 36 meets the co-ordinate axes x and y at A and B. respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is :-

A tangent to the hyperbola x^(2)-2y^(2)=4 meets x-axis at P and y-aixs at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is origin).Find the locus of R.

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.

P is a point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , and N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to

The tangent at a point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 passes through the point (0,-b) and the normal at P passes through the point (2asqrt(2),0) . Then the eccentricity of the hyperbola 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