The line `y=mx+x` and the circle `x^(2)+y^(2)=a^(2)` intersect at A and B. If `AB=2lambda`, then show that : `c^(2)=(1+m^(2))(a^(2)-lambda^(2))`.

The line y=mx+c intersects the circle x^(2)+y^(2)=r^(2) in two distinct points if

The straight-line y= mx+ c cuts the circle x^(2) + y^(2) = a^(2) in real points if :

The line 4y - 3x + lambda =0 touches the circle x^2 + y^2 - 4x - 8y - 5 = 0 then lambda=

If the line y=3x+lambda touches the hyperbola 9x^(2)-5y^(2)=45 , then the value of lambda is

Find the conditions that the line (i) y = mx + c may touch the circle x^(2) +y^(2) = a^(2) , (ii) y = mx + c may touch the circle x^(2) + y^(2) + 2gx + 2fy + c = 0 .

If the straight line y=mx is outside the circle x^(2)+y^(2)-20y+90=0 , then show that |m|lt3 .

Tangents are drawn to circle x^(2)+y^(2)=1 at its iontersection points (distinct) with the circle x^(2)+y^(2)+(lambda-3)x+(2lambda+2)y+2=0 . The locus of intersection of tangents is a straight line, then the slope of that straight line is .

The intercept on the line y=x by the circle x^(2)+y^(2)-2x=0 is AB. Show that the equation of the circle with AB as diameter is x^(2)+y^(2)-x-y=0 .

If the line y=3x+lambda touches the hyperbola 9x^(2)-5y^(2)=45 , then lambda =

Prove that the ellipse x^2/a^2 + y^2/b^2 = 1 and the circle x^2 + y^2 = ab intersect at an angle tan^(-1) (|a-b|/sqrt(ab)) .