Find the equation to the pair of lines joining the origin to the intersection of the straight line y = mx + c and the curve x2 + y2 = a2. Prove that they are at right angles if 2c2 = a2(1 + m2)
Ans: (c2 + a2m2) x2 + 2a2mxy + (c2 – a2)y2 = 0
Show that the straight lines x2(tan2θ + cos2θ) - 2xy tanθ + sin2θ = 0 make with x-axis angles such that the difference of their tangents is 2.
Prove that the straight lines joining the origin to the point of intersection of the line bx + ay - ab = 0 and the curve x2 + y2 = c2 are right angels if \(\frac1{\mathrm a^2}+\frac1{\mathrm b^2}=\frac2{\mathrm c^2}\)
Show that the straight lines joining the origin to the points of intersection of the line kx + hy = 2hk with the curve (x – h)2 + (y – k)2 = c2 are at right angles if h2 + k2 = c2.
Find the condition so that the straight lines joining the origin to the points of intersection of the line kx + hy = 2hk with the circle (x – h)2 + (y - k)2 = c2 are at right angle.
Find the equation of the lines which are right angles to the lines represented by ax2 + 2hxy + by2 = 0.
Find the single equation of the lines through the origin and perpendicular to the lines represented by the equation ax2 + 2hxy + by2 = 0.
Ans: ay2 + 2hxy + bx2 = 0
Find the equations of the two lines represented by x2 + 6xy + 9y2 + 4x + 12y - 5 = 0. Prove that the two lines are parallel.
Ans: x + 3y – 1 = 0; x + 3y + 5 = 0
Determine the two straight lines represented by 6x2 - xy - 12y2 - 8x + 29y - 14 = 0.
Ans: 2x - 3y + 2 = 0; 3x + 4y - 7 = 0
For what value of c the lines joining the origin to the point of intersection of the line x - y + c = 0 and the curve x2 + y2 + 4x - 6y - 36 = 0 may be at right angles.
Ans: c = 9 or – 4
Show that the lines joining the points of intersection of the line x + y = 1 with the curve 4x2 + 4y2 + 4x - 2y - 5 = 0 with the origin are at right angles to each other.
