If `a x+b y=1`is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`, then `a^2-b^2`is equal to`1/(a^2e^2)`(b) `a^2e^2``b^2e^2`(d) none of theseA. `1//a^(2)e^(2)`B. `a^(2)e^(2)`C. `b^(2)e^(2)` none of theseD. none of these

If `a x+b y=1`is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`,

1.	If `a x+b y=1`is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`, then `a^2-b^2`is equal to`1/(a^2e^2)`(b) `a^2e^2``b^2e^2`(d) none of theseA. `1//a^(2)e^(2)`B. `a^(2)e^(2)`C. `b^(2)e^(2)` none of theseD. none of these
Answer» Correct Answer - A Any tangent to hyperbola is `(x)/(a) sec theta-(y)/(b) tan theta=1" (1)"` The given tangent is `ax+by=1" (2)"` Comparing (1) and (2), we have `sec theta=a^(2)and tan theta=-b^(2)` Eliminating `theta`, we have `a^(4)-b^(4)=1` `"or "(a^(2)-b^(2))(a^(2)+b^(2))=1` Also, `a^(2)+b^(2)=a^(2)e^(2)` `"or "a^(2)-b^(2)=(1)/(a^(2)e^(2))`

Discussion

No Comment Found

Related InterviewSolutions

If `4(x-sqrt2)^(2)+lambda(y-sqrt3)^(2)=45 and (x-sqrt2)^(2)-4(y-sqrt3)^(2)=5` cut orthogonally, then integral value of `lambda` is ________.
Suppose the circle having equation `x^2+y^2=3`intersects the rectangular hyperbola `x y=1`at points `A ,B ,C ,a n dDdot`The equation `x^2+y^2-3+lambda(x y-1)=0,lambda in R ,`represents.a pair of lines through the origin for `lambda=-3`an ellipse through `A ,B ,C ,a n dD`for `lambda=-3`a parabola through `A , B , C ,a n dD`for `lambda=-3`a circle for any `lambda in R`A. a pair of lines through the origin for `lambda=-3`B. an ellipse through A, B, C and D for `lambda=-3`C. a parabola through A, B, C and D for `lambda=-3`D. a circle for any `lambda in R`
The equation of that chord of hyperbola `25x^(2)-16y = 400`, whose mid point is (5,3) isA. `115x - 117y = 17`B. `125x - 48y = 481`C. `127x + 33y = 341`D. `15x - 121y = 105`
The equation of the chord joining two points `(x_(1),y_(1))` and `(x_(2),y_(2))` on the rectangular hyperbola `xy=c^(2)`, isA. `(x)/(x_(1)+x_(2))+(y)/(y_(1)+y_(2))=1`B. `(x)/(x_(1)-x_(2))+(y)/(y_(1)-y_(2))=1`C. `(x)/(y_(1)+y_(2))+(y)/(x_(1)+x_(2))=1`D. `(x)/(y_(1)-y_(2))+(y)/(x_(1)-x_(2))=1`
If the chord joining the points `(a sectheta_1, b tantheta_1)` and `(a sectheta_2, b tantheta_2)` on the hyperbola `x^2/a^2-y^2/b^2=1` is a focal chord, then prove that `tan(theta_1/2)tan(theta_2/2)+(ke-1)/(ke+1)=0`, where `k=+-1`A. `(1-e)/(1+e)`B. `(e-1)/(e+1)`C. `(e+1)/(e-1)`D. `(1+e)/(1-e)`
If a chord joining `P(a sec theta, a tan theta), Q(a sec alpha, a tan alpha)` on the hyperbola `x^(2)-y^(2) =a^(2)` is the normal at P, then `tan alpha =`A. `tan theta (4 sec^(2) theta+1)`B. `tan theta (4 sec^(2) theta -1)`C. `tan theta (2 sec^(2) theta -1)`D. `tan theta (1-2 sec^(2) theta)`
Prove that any hyperbola and its conjugate hyperbola cannot have common normal.
The number of normal (s) of a rectangular hyperbola which can touch its conjugate is equal to
Normal are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`at point `theta_1a n dtheta_2`meeting the conjugate axis at `G_1a n dG_2,`respectively. If `theta_1+theta_2=pi/2,`prove that `C G_1dotC G_2=(a^2e^4)/(e^2-1)`, where `C`is the center of the hyperbola and `e`is the eccentricity.
There exist two points `P` and `Q` on the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` such that `PObotOQ`, where `O` is the origin, then the number of points in the `xy`-plane from where pair of perpendicular tangents can be drawn to the hyperbola , isA. `0`B. `1`C. `2`D. infinite

If `a x+b y=1`is tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1`, then `a^2-b^2`is equal to`1/(a^2e^2)`(b) `a^2e^2``b^2e^2`(d) none of theseA. `1//a^(2)e^(2)`B. `a^(2)e^(2)`C. `b^(2)e^(2)` none of theseD. none of these

Discussion

No Comment Found

Related InterviewSolutions

Reply to Comment