P is the mid-point of QR.
∴ ∠OPQ = 90°(Perpendicular from the centre of the circle to the chord bisects the chord)
PQ = PR = 1 cm
Also, M is the mid-point of CD.
∴∠OMP = 90°(Perpendicular from the centre of the circle to the chord bisects the chord)
In right ΔOPQ
OP=√OQ2−PQ2=√72−12=√48cmIn right ΔOMP
OM=√OP2−MP2=√48−24=√24cm
∴ OM = MP
⇒ ∠OPM = ∠POM = 45°
∴ ∠QPM = 90° - 45° = 45°
⇒∠QPD = 180° - 45° = 135°(Linear pair)
Hence, statement 1 is correct.
QR and CD are two chords of the circle that intersect at P.
∴ CP × PD = QP × PR ⇒mn =1
So, m and n are the roots of the quadratic equation
x2−10x+1=0,as the product of the roots is 1.
Hence, statement 2 is correct.
Area of ΔOPR
=×PR×OP=×1×√48=2√3cm2Area of ΔOMP
=×MP×OM=×√24×√24=12cm2Therefore, ratio of the area of ΔOPR to the area of ΔOMP
===1:2√3Hence, statement 3 is incorrect.