Straight lines `y=m x+c_1` and `y=m x+c_2` where `m in R^+,` meet the x-axis at `A_1a n dA_2,` respectively, and the y-axis at `B_1a n dB_2,` respectively. It is given that points `A_1,A_2,B_1,` and `B_2` are concylic. Find the locus of the intersection of lines `A_1B_2` and `A_2B_1` .

The line L_1-=4x+3y-12=0 intersects the x-and y-axies at Aa n dB , respectively. A variable line perpendicular to L_1 intersects the x- and the y-axis at P and Q , respectively. Then the locus of the circumcenter of triangle A B Q is

A B is a double ordinate of the parabola y^2=4a xdot Tangents drawn to the parabola at Aa n dB meet the y-axis at A_1a n dB_1 , respectively. If the area of trapezium A_1A_1B_1B is equal to 12 a^2, then the angle subtended by A_1B_1 at the focus of the parabola is equal to

Two variable chords A Ba n dB C of a circle x^2+y^2=r^2 are such that A B=B C=r . Find the locus of the point of intersection of tangents at Aa n dCdot

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 equations of the perpendicular bisectors of the sides A Ba n dA C of triangle A B C are x-y+5=0 and x+2y=0 , respectively. If the point A is (1,-2) , then find the equation of the line B Cdot

The straight line 4x + 3y - 12 = 0 intersects the x -and y-axis A and B respectively. Find the coordinates of the point at which the straight line bar(AB) is divided internally into the ratio 2 : 1.

Two straight lines (y-b)=m_1(x+a) and (y-b)=m_2(x+a) are the tangents of y^2=4a xdot Prove m_1m_2=-1.

A straight line AB intersects the y-axis at B and the perpendicular to AB at B intersects the x-axis at C . If the equation of the straight line AB is (x)/(3)-(y)/(4)=-1 find the coordinates of the points C.

If a_1, a_2 are the coefficients of x^n in the expansion of (1+x)^(2n) & (1+x)^(2n-1) respectively then a_1:a_2 will be

If the points ( a_1,b_1),(a_2,b_2) and (a_1+a_2,b_1+b_2) are collinear ,show that a_1b_2=a_2b_1 .