A tangent to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` cuts the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` at `Pa n dQ` . Show that the locus of the midpoint of `P Q` is `((x^2)/(a^2)+(y^2)/(b^2))^2=(x^2)/(a^2)-(y^2)/(b^2)dot`

If hyperbola (x^2)/(b^2)-(y^2)/(a^2)=1 passes through the focus of ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 , then find the eccentricity of hyperbola.

For the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 and (x^(2))/(b^(2))+(y^(2))/(a^(2)) =1

Tangents at right angle are drawn to the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 . Show that the focus of the middle points of the chord of contact is the curve (x^(2)/a^(2)+y^(2)/b^(2))^(2)=(x^(2)+y^(2))/(a^(2)+b^(2)) .

The area of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is

Find the equations to the common tangents to the two hyperbolas (x^2)/(a^2)-(y^2)/(b^2)=1 and (y^2)/(a^2)-(x^2)/(b^2)=1

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 a point (x_1,y_1) lies in the shaded region (x^2)/(a^2)-(y^2)/(b^2)=1 , shown in the figure, then (x^2)/(a^2)-(y^2)/(b^2)<0 Statement 2 : If P(x_1,y_1) lies outside the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 , then (x1 2)/(a^2)-(y1 2)/(b^2)<1

Statement 1 : If from any point P(x_1, y_1) on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 , tangents are drawn to the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1, then the corresponding chord of contact lies on an other branch of the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=-1 Statement 2 : From any point outside the hyperbola, two tangents can be drawn to the hyperbola.

The line y=x+rho (p is parameter) cuts the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 at P and Q, then prove that mid point of PQ lies on a^(2)y=-b^(2)x .

Length of common tangents to the hyperbolas x^2/a^2-y^2/b^2=1 and y^2/a^2-x^2/b^2=1 is