Consider the chords of the parabola `y^(2)=4x` which touches the hyperbola `x^(2)-y^(2)=1`, the locus of the point of intersection of tangents drawn to the parabola at the extremitites of such chords is a conic section having latursrectum `lambda`, the value of `lambda`, is

Options

A. The eccentricity of ellipse E equals cos `60^(@)`.
B. The foci of ellipse E are `(0,+-sqrt(3))`.
C. The length of latus-rectum of ellipse E equals 1.
D. The distance between vertices of ellipse E equals 4.

Explanation

Any tangent to given parabola is
`x.sec theta - y . Tan theta = 1 " " ...(1)`
Let `P(h,k)` be the point of intersection of tangents to the parabola `y^(2)=4x`,
so equation of chord of contact of (h, k) is ky 2(x+h) ...(2)
As, (1) and (2) are identical, so
`(sec theta)/(2)= (tan theta)/(k)=(-1)/(2h)` (on comparing)
`rArr sec theta = (-1)/(h)` and `tan theta=(k)/(2h)`
As, `sec^(2) theta - tan^(2)theta = 1`
`rArr (1)/(h^(2))-(k^(2))/(4h^(2))=1 rArr 4-k^(2)=4h^(2)`
`:. ` Locus of P(h,k) is
`E : 4x^(2)+y^(2)=4`, where is an ellipse or `(x^(2))/(1)+(y^(2))/(4)=1 " " ...(3)`
(i) As , `e^(2)=1-(1)/(4)=(3)/(4)rArr e= (sqrt(3))/(2)= sin 60^(@) rArr ` (A) is incorrect.
The foci of ellipse are `(0, +- "be" ) 3 i.e., (0, +- sqrt(3))rArr` (B) is correct.
Also, `l(L.R)=(2a^(2))/(b)=(2xx(1)^(2))/(2)=1 rArr ` (C) is correct.
Note, that distance between vertices of ellipse E = distance between (0,-2) and (0,2)=4 (D) is correct.
(ii) The circle described on vertices of ellipse E as diameter is `x^(2)+y^(2)=4 " " ....(4)`
as, circle in equation (4) intersects orthogonally the circle `x^(2)+y^(2)-4x-2y+k^(2)=0`,
so using condition of orthogonality, we get `0=k^(2)-4 rArr k = +-2`