If the lines `L_1a n dL_2` are tangents to `4x^2-4x-24 y+49=0` and are normals for `x^2+y^2=72 ,` then find the slopes of `L_1` and `L_2dot`

The combined equation of the lines l_1a n dl_2 is 2x^2+6x y+y^2=0 and that of the lines m_1a n dm_2 is 4x^2+18 x y+y^2=0 . If the angle between l_1 and m_2 is alpha then the angle between l_2a n dm_1 will be

The combined equation of the lines L_1 and L_2 is 2x^2+6xy+y^2=0 , and that of the lines L_3 and L_4 is 4x^2+18xy+y^2=0 . If the angle between L_1 and L_4 be alpha , then the angle between L_1 and L_3 will be .

Let l_(1) and l_(2) be the two lines which are normal to y^(2)=4x and tangent to x^(2)=-12y respectively (where, l_(1) and l_(2) are not the x - axis). Then, the product of the slopes of l_(1)and l_(2) is

Find the equation of tangents to hyperbola x^(2)-y^(2)-4x-2y=0 having slope 2.

The combined equation of the lines L_(1) and L_(2) is 2x^(2)+6xy+y^(2)=0 and that lines L_(3) and L_(4) is 4x^(2)+18xy+y^(2)=0 . If the angle between L_(1) and L_(4) be alpha , then the angle between L_(2) and L_(3) will be

Let the lines l_(1) and l_(2) be normals to y^(2)=4x and tangents to x^(2)=-12y (where l_(1) and l_(2) are not x - axis). The absolute value of the difference of slopes of l_(1) and l_(2) is

Consider two lines L_1a n dL_2 given by a_1x+b_1y+c_1=0a n da_2x+b_2y+c_2=0 respectively where c1 and c2 !=0, intersecting at point PdotA line L_3 is drawn through the origin meeting the lines L_1a n dL_2 at Aa n dB , respectively, such that PA=P B . Similarly, one more line L_4 is drawn through the origin meeting the lines L_1a n dL_2 at A_1a n dB_2, respectively, such that P A_1=P B_1dot Obtain the combined equation of lines L_3a n dL_4dot

The portion of the line 4x+5y=20 in the first quadrant is trisected by the lines L_(1) and L_(2) passing through the origin.The tangent of an angle between the lines L_(1) and L_(2) is:

Two lines L_(1) and L_(2) of slops 1 are tangents to y^(2)=4x and x^(2)+2y6(2)=4 respectively, such that the distance d units between L_(1) and L _(2) is minimum, then the value of d is equal to