At the midpoints of line segment AB which is p units long, a perpendicular MR is erected with length q - Maths - Quadratic Equations

Question

At the midpoints of line segment AB which is p units long, a
perpendicular MR is erected with length q units An arc is described
from R with a radius equalt to 1/2 AB, meeting AB at T Then AT
& TB are roots of which equation ?

Manbar Singh · Accepted Answer

We have, AB = p units
Now, M is the mid point of AB, then AM = MB = p2 unitsNow, RM⊥AB and RM = q units
Now, RT = p2 unitsIn ∆RMT,MT2 = RT2 - RM2   Pythagoras theorem⇒MT2 = p22 - q2 = p24-q2 = p2-4q24Now, MT = p2-4q22Now, AT = AM+MT = p2+p2-4q22 = p+p2-4q22TB = AB-AT = p- p+p2-4q22 = p-p2-4q22Now, AT + TB = p+p2-4q22 + p-p2-4q22 = pAT × BT = p+p2-4q22 × p-p2-4q22 = p2-p2+4q24 = q2Now, required quadratic equation is,     x2 - AT+BTx + AT×BT = 0⇒x2 - px + q2 = 0

At the midpoints of line segment AB which is p units long, a perpendicular MR is erected with length q units. An arc is described from R with a radius equalt to 1/2 AB, meeting AB at T. Then AT & TB are roots of which equation ?