If `y=mx+c` is the tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` at any point on it, show that `c^(2)=a^(2)m^(2)+b^(2).` Find the coordinates of the point of contact.

The slop of the tangent to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 at the point (a cos theta, b sin theta) - is

Find the slope of normal to the ellipse , (x^(2))/(a^(2))+(y^(2))/(b^(2))=2 at the point (a,b).

The slope of the tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point ( x_(1),y_(1)) is-

Find the equation of the tangent to the ellipse x^2/a^2+y^2/b^2=1 at (x= 1,y= 1) .

If any tangent to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 intercepts equal lengths l on the axes, then find l .

If the tangent at any point of the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 makes an angle alpha with the major axis and an angle beta with the focal radius of the point of contact, then show that the eccentricity of the ellipse is given by e=cosbeta/(cosalpha)

Find the equations of the tangent and normal to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 at the point (x0, y0).

If the tangents to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 make angles alphaa n dbeta with the major axis such that tanalpha+tanbeta=gamma, then the locus of their point of intersection is (a) x^2+y^2=a^2 (b) x^2+y^2=b^2 (c) x^2-a^2=2lambdax y (d) lambda(x^2-a^2)=2x y

Show that the straight line (x)/(a)+(y)/(b)=2 touches the curve ((x)/(a))^(3)+((y)/(b))^(3)=2 , find the coordinates of the point of contact.

If the tangent to the parabola y= x^(2) + 6 at the point (1, 7) also touches the circle x^(2) + y^(2) + 16x + 12y + c =0 , then the coordinates of the point of contact are-