`C` is the center of the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` The tangent at any point `P` on this hyperbola meet the straight lines `b x-a y=0` and `b x+a y=0` at points `Qa n dR` , respectively. Then prove that `C QdotC R=a^2+b^2dot`

The number of normals to the hyperbola x^(2)/a^(2) - y^(2)/b^(2) = 1 from an external point is _______

The tangent at P on the hyperbola (x^(2))/(a^(2)) -(y^(2))/(b^(2))=1 meets one of the asymptote in Q. Then the locus of the mid-point of PQ is

If any line perpendicular to the transverse axis cuts the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and the conjugate hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 at points Pa n dQ , respectively, then prove that normal at Pa n dQ meet on the x-axis.

Find the equations of the tangent and normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point (x_(0), y_(0)).

Let 'p' be the perpendicular distance from the centre C of the hyperbola x^2/a^2-y^2/b^2=1 to the tangent drawn at a point R on the hyperbola. If S & S' are the two foci of the hyperbola, then show that (RS + RS')^2 = 4 a^2(1+b^2/p^2) .

P is a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transvers axis at Tdot If O is the center of the hyperbola, then find the value of O TxO Ndot

Tangents are drawn to the hyperbola x^(2)/9 - y^(2)/4 parallel to the straight line 2x - y= 1 . One of the points of contact of tangents on the hyperbola is

Prove that the length of the latusrection of the hyperbola x^(2)/a^(2) -y^(2)/b^(2) =1 " is" (2b^(2))/a

If the tangents to the parabola y^2=4a x intersect the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 at Aa n dB , then find the locus of the point of intersection of the tangents at Aa n dBdot

A tangent is drawn at any point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2)) =1 . If this tangent is intersected by the tangents at the vertices at points P and Q, then which of the following is/are true