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Text Solution

Solution :

Let locus of point `P(h,k)`
`Eqn.` of tangent from point `P` to the hyperbola is:
`y = mx pm sqrt(a^2*m^2 - a^2)`
`y = mx pm a*sqrt(m^2 -1)`
`Eqn.` of normal to this hyperbola:
`a^2*y_1(x-x_1) + a^2*x_1(y-y_1)`
Tangents starts from point `P(h,k)` , so it is point of contact. And since this point is normal to the curve, it satisfies `Eqn.` of normal.
`a^2*k(x-h) + a^2*h(y-k) = 0`
Converting it to slope form:
`kx - kh + hy - kh = 0`
`y = -k/h*x + 2k`
And from equation of tangent:
`k = mh pm a*sqrt(m^2-1)`
`rArr m = -k/h` ......`(eq.1)`
and,
`a*sqrt(m^2-1) = 2k`
Squaring both sides
`a^2*(m^2-1) = 4k^2`
`a^2*((-k/h)^2 - 1) = 4k^2`
`k^2/h^2 - 1 = 4k^2/a^2`
Divinding above `Eqn.` by `k^2`
`1/h^2 -1/k^2 = 4/a^2`
`:.` locus of point P(h,k) is
`1/x^2 -1/y^2 = 4/a^2`**Analytical definition of hyperbola**

**Equation of hyperbola in standard form**

**Various results of hyperbola**

**Rectangular or equilateral Hyperbola**

**Difference of Focal radii of any point is equal to the length of major axis**

**Position of a point with respect to hyperbola**

**Conjugate hyperbola**

**Comparison of Hyperbola and its conjugate hyperbola**

**Auxiliary Circle and eccentric angle**

**Relation between equation of Hyperbola and major minor axis when axis is not parallel to co ordinate axes**