3 free questions from this chapter (the full set has 30). Marking: Multi-correct: +4 all correct, +1 per correct option if none wrong, −2 if any wrong · Numerical: +4 / 0 · Matching: +3 / −1.
Question 1Level 2 · ExamSingle correct
A variable chord PQ of the circle x2+y2=25 subtends a right angle at the point A(3,0), that is, ∠PAQ=90∘. The midpoints M of all such chords fill a circle. The radius of this circle is ______. Exactly one option is correct.
A41
B241
C234
D5
Show answer and solution
Answer: B
Let M(h,k) be the midpoint of PQ. Then OM⊥PQ and MP2=25−OM2=25−h2−k2.
∠PAQ=90∘ means A lies on the circle with diameter PQ, i.e. MA=MP. Hence
(h−3)2+k2=25−h2−k2⟺2h2+2k2−6h−16=0⟺h2+k2−3h−8=0.
This circle has centre (23,0) and radius2=49+8=441.
Converse: every point of it is at distance at most 23+241<5 from O and is not O, so it is the midpoint of a chord perpendicular to OM; for that chord MA=MP, so ∠PAQ=90∘ (A is inside the circle, so A=P,Q). The locus is the whole circle.
Radius =241.
Trap: forgetting to divide 2h2+2k2−6h−16=0 by 2 before reading off the radius gives 5.
Difficulty: Hard
Question 2Level 2 · ExamSingle correct
From the point P(5,1) the two tangents PT1 and PT2 are drawn to the circle (x−1)2+(y+2)2=9, touching it at T1 and T2. The area of triangle PT1T2 is ______. Exactly one option is correct.
A12
B2596
C25144
D25192
Show answer and solution
Answer: D
Centre C(1,−2), r=3, CP=16+9=5, so the tangent length is L=25−9=4.
Let φ=∠CPT1; then sinφ=53, cosφ=54.
The chord of contact is perpendicular to CP; its half-length is Lsinφ=512 and its distance from P is Lcosφ=516.
Area =21⋅524⋅516=25192 (equivalently r2+L2rL3).
Trap: 12=rL is the area of the quadrilateral CT1PT2, not of the triangle.
Difficulty: Hard
Question 3Level 2 · ExamOne or more correct
Let C1 and C2 be the two circles that touch both coordinate axes and pass through the point (2,1), the radius of C1 being smaller than that of C2. Which of the following statements is/are correct? One or more options may be correct.
AThe sum of the radii of C1 and C2 is 6.
BC1 and C2 cut each other orthogonally.
CThe common chord of C1 and C2 lies on the line x+y=3.
DC1 and C2 have exactly two common tangents.
Show answer and solution
Answer: A, C, D
A circle touching both axes lies in one quadrant; since (2,1) is in the first quadrant, the centre is (r,r), r>0.
(2−r)2+(1−r)2=r2⟺r2−6r+5=0⟺r=1 or 5.
C1:x2+y2−2x−2y+1=0 (centre (1,1), r1=1); C2:x2+y2−10x−10y+25=0 (centre (5,5), r2=5).
(A) r1+r2=6. True.
(B) d2=32 while r12+r22=26, so not orthogonal. False.
(C) C1−C2:8x+8y−24=0, i.e. x+y=3 (it passes through (2,1) and (1,2)). True.
(D) d=42≈5.66 and r2−r1=4<d<6=r1+r2: the circles cut at two points, so they have exactly two common tangents. True.
Difficulty: Hard
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