Prove that any hyperbola and its conjugate hyperbola cannot have common normal.
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Consider hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1.` Equation of normal to hyperbola at point `P(a sec theta, b tan theta)` is `ax cos theta+by cot theta=a^(2)+b^(2)" (1)"` Equation of normal to hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=-1` at point `Q( a tan phi, b sec phi)` is `ax cot phi+"by" cos phi=a^(2)+b^(2)" (2)"` If Eqs. (1) and (2) represent the same straight line, then `(cot phi)/(cos theta)=(cos phi)/(cot theta)=1` `rArr" "tan phi = sec theta and sec phi = tan theta` `rArr" "sec^(2)phi-tan^(2)phi=tan^(2)theta-sec^(2)theta=-1,` which is not possible. Thus, hyperbola and its conjugate hyperbola cannot have common normal.
If a variable line has its intercepts on the coordinate axes e and e^(prime), where e/2a n d e^(prime)/2 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle x^2+y^2=r^2, where r=
Column I|Column II (a)Two intersecting circle| p. have a common tangent (b)Two mutually external circles| q. have a common normal (c)Two circles, one strictly inside the other| r. do not have a common tangent (d)Two branches of a hyperbola| s.do not have a common normal
e_(1) and e_(2) are respectively the eccentricites of a hyperbola and its conjugate. Prove that (1)/(e_1^(2))+(1)/(e_2^2) =1
Find the equation of the normal to the hyperbola 3x^(2)-4y^(2)=12 at the point (x_(1),y_(1)) on it. Hence, show that the straight line x+y+7=0 is a normal to the hyperbola. Find the coordinates of the foot of the normal.
Find the eccentricity of a hyperbola whose conjugate axis and latus rectum are equal.
Find the eccentricity of a hyperbola whose conjugate axis and latus rectum are equal.
If e_1 and e_2 be the eccentricities of a hyperbola and its conjugate, show that 1/(e_1^2)+1/(e_2^2)=1 .
e_1 and e_2 are respectively the eccentricities of a hyperbola and its conjugate.Prove that 1/e_1^2+1/e_2^2=1
A variable chord of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,(b > a), subtends a right angle at the center of the hyperbola if this chord touches. a fixed circle concentric with the hyperbola a fixed ellipse concentric with the hyperbola a fixed hyperbola concentric with the hyperbola a fixed parabola having vertex at (0, 0).
If asymptotes of hyperbola bisect the angles between the transverse axis and conjugate axis of hyperbola, then what is eccentricity of hyperbola?
