Match the following lists:

1.	Match the following lists:
Answer» (a) Tangent to the ellipe at `P(phi)` is `(x)/(4) cos phi+(y)/(2) sin phi=1` It must pass through the center of the circle. Hence, `(4)/(4) cos phi+(2)/(2)sin phi=1` `or cos phi+sin phi=1` or `1+sin 2 phi phi=1` or `sin 2phi=0` i.e.,` 2phi=0 or pi` i.e., `(phi)/(2)=0 or (pi)/(4)` (b) Consider and point `P(sqrt(6) cos, theta, sqrt(2) sin theta)` on the ellipe `(x^(2))/(6)+(y^(2))/(2)=1` Given thast P=2. Therefore, `6 cos^(2) theta+2sin^(2) theta=0` or `4cos^(2) theta=2` or `cos theta=+-(1)/(sqrt(2))` i.e.,` theta=(pi)/(4) or (5pi)/(4)` (c) Sovling the equation of ellipse and parbola (eliminating `(x^(2)`)n we have `y-1+4y^(2)=4` or `4y^(2)+y-5=0` `(4y+5)(y-1)=0` or y=1, x=0` The curves touch at (0,1) so, the angle of intersection is 0. (d) The normal at`P(a cos theta, b sin theta)` is `(ax)/(cos thet)-(hx)/(sin theta)=a^(2)-b^(2)` Where `a^(2)=14,b^(2)=5` It means the cruve again at Q `(2 theta)`, i.e., `(a cos 2 theta, b sin 2theta)` Hecen, `(a)/(costheta)a cos 2theta-(b)/(sin theta)(b 2sintheta)=a^(2)-b^(2)` `or (14)/(cos theta) cos 2theta-(5)/(sin theta)(sin 2 theta)=14-5` or `28 cos^(2)theta-14-10 sec^(2)theta=9 cos theta` `or 18 cos^(2)theta-9cos theta-14=0` `or (6 cos theta-7)(3 cos theta+2)` or `cos theta=-(2)/(3)`

Discussion

No Comment Found

Related InterviewSolutions

The number of points on the ellipse `(x^2)/(50)+(y^2)/(20)=1`from which a pair of perpendicular tangents is drawn to the ellipse `(x^2)/(16)+(y^2)/9=1`is0 (b) 2(c) 1 (d)4
Tangents are drawn from the points on the line x-y-5=0 ot `x^(2)+4y^(2)=4` . Prove that all the chords of contanct pass through a fixed point
If radius of the director circle of the ellipse `((3x+4y-2)^(2))/(100)+((4x-3y+5)^(2))/(625) =1` isA. 6B. `sqrt(34)`C. `sqrt(29)`D. `sqrt(26)`
PQ and QR are two focal chords of an ellipse and the eccentric angles of P,Q,R are `2alpha, 2beta, 2 gamma`, respectively then `tan beta gamma` is equal toA. `cot alpha`B. `cot^(2)alpha`C. `2 cot alpha`D. None of these
If the chords of contact of tangents from twopoinst `(x_1, y_1)`and `(x_2, y_2)`to theellipse `(x^2)/(a^2)+(y^2)/(b^2)=1`are atright angles, then find the value of `(x_1x_2)/(y_1y_2)dot`
Tangents are drawn from any point on the circle `x^(2)+y^(2) = 41` to the ellipse `(x^(2))/(25)+(y^(2))/(16) =1` then the angle between the two tangents isA. `(pi)/(4)`B. `(pi)/(3)`C. `(pi)/(6)`D. `(pi)/(2)`
Prove that the chords of constant of perpendicular tangents to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` touch another fixed ellipse `(x^(2))/(a^(4))+(y^(2))/(b^(4))=(1)/((a^(2)+b^(2)))`
Prove that the chords of contact of pairs of perpendicular tangents to the ellipse `x^2/a^2+y^2/b^2=1` touch another fixed ellipse.A. `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =(1)/((2a^(2)+b^(2)))`B. `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =(2)/((a^(2)-b^(2)))`C. `(x^(2))/(a^(4))+(y^(2))/(b^(4)) =(1)/((a^(2)+b^(2)))`D. `(x^(2))/(a^(2))-(y^(2))/(b^(2)) =(2)/((3a^(2)-b^(2)))`
Find the equation of the ellipse (referred to itsaxes as the axes of `xa n dy`, respectively) whose foci are `(+-2,0)`andeccentricity is `1/2`
Find the equation of the ellipse, with major axis along the x-axis and passing through the points `(4, 3)`and `(-1, 4)`.

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment