Chapter 7: Conic Sections and Their Geometry (Set-5)
The circle x2+y2−6x−8y+9=0x2+y2−6x−8y+9=0 has director circle equation
A x2+y2−6x−8y+0=0x2+y2−6x−8y+0=0
B x2+y2+6x+8y+25=0x2+y2+6x+8y+25=0
C x2+y2−6x−8y+25=0x2+y2−6x−8y+25=0
D x2+y2−6x−8y−25=0x2+y2−6x−8y−25=0
For circle x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0, director circle is x2+y2+2gx+2fy+g2+f2=0x2+y2+2gx+2fy+g2+f2=0. Here 2g=−6⇒g=−32g=−6⇒g=−3, 2f=−8⇒f=−42f=−8⇒f=−4. So add g2+f2=25g2+f2=25.
For circle x2+y2=16×2+y2=16, polar of point P(5,0)P(5,0) is
A x=16/5x=16/5
B x=5/16x=5/16
C y=16/5y=16/5
D y=5/16y=5/16
For x2+y2=r2x2+y2=r2, polar of (x1,y1)(x1,y1) is xx1+yy1=r2xx1+yy1=r2. With (5,0)(5,0) and r2=16r2=16: 5x=16⇒x=16/55x=16⇒x=16/5. It’s the chord of contact from PP.
If polar of PP w.r.t x2+y2=25×2+y2=25 is 3x+4y=253x+4y=25, then PP is
A (4,3)(4,3)
B (3,4)(3,4)
C (5,0)(5,0)
D (0,5)(0,5)
Polar of (x1,y1)(x1,y1) is xx1+yy1=25xx1+yy1=25. Comparing with 3x+4y=253x+4y=25 gives x1=3,y1=4×1=3,y1=4. So point is (3,4)(3,4). It lies outside since 32+42=2532+42=25 actually lies on circle.
For circle x2+y2=25×2+y2=25, point P(3,4)P(3,4) gives number of tangents
A Two tangents
B No tangent
C Infinite tangents
D One tangent
P(3,4)P(3,4) satisfies 32+42=2532+42=25, so it lies on the circle. From a point on the circle, exactly one tangent can be drawn at that point. Tangent is perpendicular to radius OPOP.
Common tangents of two circles touch externally when center distance dd equals
A r1+r2r1+r2
B ∣r1−r2∣∣r1−r2∣
C r1r2r1r2
D r12+r22r12+r22
External tangency means circles touch at one point outside, with line of centers passing through contact. The distance between centers equals sum of radii. Then there are three common tangents (two direct, one at contact).
For two circles, radical axis passes through their intersection points because
A Radii equal there
B Tangents equal there
C Powers equal there
D Centers equal there
Intersection points lie on both circles, so their power w.r.t each circle is zero. Hence powers are equal at intersection points. Therefore those points satisfy the radical axis equation S1−S2=0S1−S2=0.
A chord of x2+y2=r2x2+y2=r2 subtends right angle at center iff its distance from center is
A r/2r/2
B r/2r/2
C r2r2
D 00
If chord subtends 90∘90∘ at center, then half-angle is 45∘45∘. Distance from center to chord is rcos45∘=r/2rcos45∘=r/2. Chord length becomes r2r2, consistent with geometry.
The locus of midpoints of chords of circle x2+y2=r2x2+y2=r2 that pass through fixed point PP is a
A Circle
B Ellipse
C Hyperbola
D Line
Midpoints of chords through a fixed point lie on the circle’s “midpoint line” perpendicular to the line joining the center and the fixed point. This is a standard chord-midpoint locus result in circle geometry.
If a circle touches both coordinate axes and passes through (4,3)(4,3), its radius is
A 11
B 22
C 33
D 44
Touching both axes means center (r,r)(r,r) in first quadrant and radius rr. Passing through (4,3)(4,3) gives (4−r)2+(3−r)2=r2(4−r)2+(3−r)2=r2. Expanding: 25−14r+2r2=r2⇒r2−14r+25=0⇒r=7±2625−14r+2r2=r2⇒r2−14r+25=0⇒r=7±26. None matches options.
The correct radius is r=7±26r=7±26. Since neither value is among the options, this MCQ cannot be correct as written.
Circle x2+y2+2x+2y+1=0x2+y2+2x+2y+1=0 represents
A Point circle
B Real circle
C Imaginary circle
D Two circles
Complete squares: (x+1)2+(y+1)2=1−1=0(x+1)2+(y+1)2=1−1=0. Radius is 0, so circle reduces to a single point (−1,−1)(−1,−1). This is called a point circle (degenerate).
For parabola y2=4axy2=4ax, equation of chord with midpoint (x1,y1)(x1,y1) is
A T=0T=0
B S=0S=0
C S1=0S1=0
D T=S1T=S1
For conic S=0S=0, chord with midpoint (x1,y1)(x1,y1) is given by T=S1T=S1, where TT is bilinear form at (x1,y1)(x1,y1) and S1S1 is value at midpoint. It’s a key chord-midpoint formula.
For parabola y2=4axy2=4ax, pair of tangents from point (h,k)(h,k) is real if
A k2>4ahk2>4ah
B k2=4ahk2=4ah
C k2<4ahk2<4ah
D h=0h=0
Point (h,k)(h,k) lies outside parabola when k2>4ahk2>4ah. From an external point, two real tangents can be drawn. On the parabola (k2=4ah)(k2=4ah) only one tangent exists; inside gives no real tangents.
The combined equation of tangents from (h,k)(h,k) to y2=4axy2=4ax is
A S+S1=0S+S1=0
B S−T=0S−T=0
C SS1=T2SS1=T2
D S1T=0S1T=0
For any conic S=0S=0, the pair of tangents from (h,k)(h,k) is given by SS1=T2SS1=T2. Here S=y2−4axS=y2−4ax. This produces a second-degree equation representing both tangents together.
For y2=4axy2=4ax, condition that tangent with slope mm exists is
A Always exists
B m≠0m=0
C m>0m>0
D m<0m<0
Any real slope mm corresponds to a tangent of the parabola y2=4axy2=4ax because tangent in slope form is y=mx+amy=mx+ma. This is valid for all nonzero mm, but vertical tangent doesn’t occur here.
Tangent in slope form for y2=4axy2=4ax is
A y=mx+amy=mx+ma
B y=mx+amy=mx+am
C y=mx+may=mx+am
D y=mx−ay=mx−a
Standard slope form tangent to parabola y2=4axy2=4ax is y=mx+amy=mx+ma. Substituting into parabola gives a quadratic with discriminant zero, ensuring tangency. Note m≠0m=0.
For parabola y2=4axy2=4ax, tangent at parameter tt meets xx-axis at
A (at2,0)(at2,0)
B (2at2,0)(2at2,0)
C (a/t2,0)(a/t2,0)
D (−at2,0)(−at2,0)
Tangent at parameter tt is ty=x+at2ty=x+at2. Put y=0y=0: 0=x+at2⇒x=−at20=x+at2⇒x=−at2. So it meets xx-axis at (−at2,0)(−at2,0). Useful intercept property.
For parabola x2=4ayx2=4ay, the pair of tangents from (h,k)(h,k) is real if
A h2=4akh2=4ak
B h2<4akh2<4ak
C h2>4akh2>4ak
D k=0k=0
For x2=4ayx2=4ay, external points satisfy h2>4akh2>4ak. Then two real tangents exist. If equality holds, point lies on parabola giving one tangent; inside region has no real tangents.
For translated parabola (y−k)2=4a(x−h)(y−k)2=4a(x−h), focus is
A (h+a,k)(h+a,k)
B (h−a,k)(h−a,k)
C (h,k+a)(h,k+a)
D (h,k−a)(h,k−a)
Translation moves vertex to (h,k)(h,k). For right-opening parabola, focus is aa units from vertex along axis direction, so focus becomes (h+a,k)(h+a,k). Directrix becomes x=h−ax=h−a.
For parabola, the locus of intersection of tangents at endpoints of a focal chord is
A Tangent axis
B Directrix
C Circle
D Vertex line
For y2=4axy2=4ax, tangents at endpoints of a focal chord intersect on the line x=−ax=−a, called the tangent at vertex (tangent axis). This is a known harder property linking focus chord to tangents.
For parabola, the director circle is
A Always real
B Always imaginary
C Not defined
D Same as directrix
“Director circle” is typically used for ellipse and hyperbola (and circle). For a parabola, there is no finite locus of points from which tangents are perpendicular in the same standard sense, so director circle isn’t defined.
For ellipse, pair of tangents from (h,k)(h,k) to x2a2+y2b2=1a2x2+b2y2=1 is
A SS1=T2SS1=T2
B S+S1=0S+S1=0
C T=0T=0
D S1=0S1=0
For any conic S=0S=0, combined equation of pair of tangents from point (h,k)(h,k) is SS1=T2SS1=T2. Here S=x2a2+y2b2−1S=a2x2+b2y2−1. It yields both tangents together.
For ellipse, tangents from a point are real if the point lies
A Inside ellipse
B On ellipse
C At center
D Outside ellipse
From a point outside ellipse, two real tangents can be drawn. From a point on ellipse, only one tangent exists at that point. From an interior point, no real tangents exist because any line meets ellipse twice.
For ellipse x2a2+y2b2=1a2x2+b2y2=1, slope form tangent is
A y=mx±a2+b2m2y=mx±a2+b2m2
B y=mx±a2−b2m2y=mx±a2−b2m2
C y=mx±a2m2+b2y=mx±a2m2+b2
D y=mx±(a+bm)y=mx±(a+bm)
Tangent with slope mm to ellipse has form y=mx±a2m2+b2y=mx±a2m2+b2. This comes from replacing yy by mx+cmx+c in ellipse and using discriminant zero for tangency.
For ellipse, equation of director circle is real when
A a2>2b2a2>2b2
B a2=2b2a2=2b2
C a2<2b2a2<2b2
D a=ba=b
Director circle for ellipse is x2+y2=a2−2b2x2+y2=a2−2b2 (major along xx). It is real only if right side is positive, i.e., a2−2b2>0a2−2b2>0.
For ellipse, chord of contact from point (h,k)(h,k) is given by
A S=0S=0
B T=0T=0
C S1=0S1=0
D SS1=0SS1=0
For conic S=0S=0, polar (chord of contact) of (h,k)(h,k) is T=0T=0. For ellipse, it becomes hxa2+kyb2=1a2hx+b2ky=1 when point lies on ellipse; otherwise gives chord of contact line.
For ellipse, the locus of intersection of tangents at endpoints of a diameter is
A The diameter itself
B The tangent at endpoints
C Conjugate diameter line
D Auxiliary circle
Tangents at endpoints of a diameter are parallel to the conjugate diameter direction. Their intersection points lie along the conjugate diameter line concept. This is part of conjugate diameter theory used in harder ellipse geometry.
If ellipse has a=5a=5, e=3/5e=3/5, then latus rectum length is
A 16/516/5
B 32/532/5
C 25/325/3
D 10/310/3
c=ae=5⋅3/5=3c=ae=5⋅3/5=3. Then b2=a2−c2=25−9=16b2=a2−c2=25−9=16. Latus rectum length =2b2/a=2⋅16/5=32/5=2b2/a=2⋅16/5=32/5.
Using e=c/ae=c/a, c=3c=3. Then b2=25−9=16b2=25−9=16. Latus rectum length is 2b2/a=2⋅16/5=32/52b2/a=2⋅16/5=32/5. This uses standard ellipse relation.
For ellipse, the polar of a point on ellipse is
A Normal at point
B Director circle
C Asymptote
D Tangent at point
If the point lies on the conic, its polar coincides with the tangent at that point. This follows because chord of contact degenerates to the contact point itself, giving exactly the tangent line.
For ellipse, sum of focal distances equals 2a2a; difference of focal distances is
A Not constant
B Constant 2a2a
C Constant 2b2b
D Constant 2c2c
Ellipse definition fixes the sum PF1+PF2=2aPF1+PF2=2a. The difference PF1−PF2PF1−PF2 changes from point to point; only in hyperbola the absolute difference is constant. This distinguishes ellipse from hyperbola.
If an ellipse has directrices x=±10x=±10 and a=8a=8, then ee is
A 3/53/5
B 5/45/4
C 4/54/5
D 8/108/10
For ellipse (major along xx), directrices are x=±a/ex=±a/e. Given a/e=10⇒e=a/10=8/10=4/5a/e=10⇒e=a/10=8/10=4/5. Since e<1e<1, it’s consistent with ellipse.
For hyperbola, slope form tangent to x2a2−y2b2=1a2x2−b2y2=1 is
A y=mx±a2m2+b2y=mx±a2m2+b2
B y=mx±a2m2−b2y=mx±a2m2−b2
C y=mx±b2−a2m2y=mx±b2−a2m2
D y=mx±(a+bm)y=mx±(a+bm)
Put y=mx+cy=mx+c in hyperbola and enforce discriminant zero. The result is c2=a2m2−b2c2=a2m2−b2. Hence tangent is y=mx±a2m2−b2y=mx±a2m2−b2. Real tangents need a2m2≥b2a2m2≥b2.
For hyperbola, condition that tangent with slope mm is real is
A a2m2≥b2a2m2≥b2
B a2m2≤b2a2m2≤b2
C a2m2=b2/2a2m2=b2/2
D Always real
From slope form y=mx±a2m2−b2y=mx±a2m2−b2, the square root must be real. So a2m2−b2≥0⇒a2m2≥b2a2m2−b2≥0⇒a2m2≥b2. This restricts slopes.
For hyperbola, pair of tangents from point (h,k)(h,k) is
A T=0T=0
B S1=0S1=0
C S+S1=0S+S1=0
D SS1=T2SS1=T2
Same general conic result: combined equation of pair of tangents from (h,k)(h,k) to conic S=0S=0 is SS1=T2SS1=T2. It applies to hyperbola as well, producing two tangents when external.
For rectangular hyperbola xy=c2xy=c2, the tangent at (x1,y1)(x1,y1) is
A xy1−x1y=c2xy1−x1y=c2
B xy=c2xy=c2
C xy1+x1y=2c2xy1+x1y=2c2
D x+y=cx+y=c
Differentiate xy=c2xy=c2: y+xdydx=0y+xdxdy=0. At (x1,y1)(x1,y1), tangent is y1(x−x1)+x1(y−y1)=0⇒xy1+x1y=2x1y1=2c2y1(x−x1)+x1(y−y1)=0⇒xy1+x1y=2x1y1=2c2.
For hyperbola, asymptotes of (x−h)2a2−(y−k)2b2=1a2(x−h)2−b2(y−k)2=1 are
A y−k=±ba(x−h)y−k=±ab(x−h)
B y−k=±ab(x−h)y−k=±ba(x−h)
C y=±baxy=±abx
D x=±abyx=±bay
Asymptotes come from setting the “1” to zero: (x−h)2a2−(y−k)2b2=0a2(x−h)2−b2(y−k)2=0. This gives y−kb=±x−haby−k=±ax−h, i.e. y−k=±ba(x−h)y−k=±ab(x−h).
Hyperbola’s conjugate hyperbola has equation
A x2a2+y2b2=1a2x2+b2y2=1
B y2a2−x2b2=−1a2y2−b2x2=−1
C x2−y2=a2x2−y2=a2
D x2a2−y2b2=−1a2x2−b2y2=−1
Conjugate hyperbola shares same center and asymptotes but changes sign of constant. For standard hyperbola, replacing 1 by −1−1 produces conjugate hyperbola. It opens along the conjugate axis direction.
For hyperbola, foci are at (±c,0)(±c,0) where
A c2=a2+b2c2=a2+b2
B c2=a2−b2c2=a2−b2
C c2=b2−a2c2=b2−a2
D c=a−bc=a−b
In hyperbola x2a2−y2b2=1a2x2−b2y2=1, focus distance satisfies c2=a2+b2c2=a2+b2. This ensures c>ac>a, hence eccentricity e=c/a>1e=c/a>1, matching hyperbola definition.
For hyperbola, the polar of point (x1,y1)(x1,y1) is
A xx1a2−yy1b2=0a2xx1−b2yy1=0
B xx1a2+yy1b2=1a2xx1+b2yy1=1
C xx1a2−yy1b2=1a2xx1−b2yy1=1
D xx1+yy1=0xx1+yy1=0
For standard hyperbola, the polar of point (x1,y1)(x1,y1) w.r.t x2a2−y2b2=1a2x2−b2y2=1 is xx1a2−yy1b2=1a2xx1−b2yy1=1. If point lies on hyperbola, it becomes tangent.
Hyperbola’s latus rectum endpoints are (c,±b2/a)(c,±b2/a); length is
A b2/ab2/a
B 2b2/a2b2/a
C 2a2/b2a2/b
D abab
At x=cx=c, substitution gives y2=b4a2y2=a2b4, so endpoints are y=±b2/ay=±b2/a. Distance between them is 2b2/a2b2/a. This matches standard latus rectum length for hyperbola.
Hyperbola’s director circle is x2+y2=a2+b2x2+y2=a2+b2; it means tangents from a point on it are
A Perpendicular
B Parallel
C Equal length
D Coincident
Director circle is defined as locus of points from which the pair of tangents to the conic are perpendicular. For standard hyperbola, it becomes x2+y2=a2+b2x2+y2=a2+b2. Any point on it yields right-angle tangents.
For cone x2+y2=z2x2+y2=z2, intersection with plane z=2z=2 is
A Circle radius 4
B Line pair
C Parabola curve
D Circle radius 2
Put z=2z=2 in x2+y2=z2x2+y2=z2 gives x2+y2=4×2+y2=4. That is a circle of radius 2 in plane z=2z=2. This shows cone cross-sections at constant height are circles.
A cone is invariant under scaling (x,y,z)→(kx,ky,kz)(x,y,z)→(kx,ky,kz) because its equation is
A Homogeneous degree 1
B Non-homogeneous
C Homogeneous degree 2
D Always linear
If equation has all terms of total degree 2, then substituting (kx,ky,kz)(kx,ky,kz) multiplies the whole equation by k2k2. So if original equals zero, scaled also equals zero. This scaling property is cone geometry.
For quadratic Ax2+By2+Cz2+2Fyz+2Gzx+2Hxy=0Ax2+By2+Cz2+2Fyz+2Gzx+2Hxy=0, it represents a real cone when
A It has real generators
B It has no axis
C It has constant term
D It has linear terms
A real cone means there exist real nonzero solutions (x,y,z)(x,y,z) satisfying the homogeneous quadratic, forming real generator lines. If the quadratic form is definite, only trivial solution exists, giving an imaginary cone.
In general conic, removing xyxy term by rotation uses angle θθ satisfying
A tan2θ=A−CBtan2θ=BA−C
B tan2θ=BA−Ctan2θ=A−CB
C tanθ=BA+Ctanθ=A+CB
D tanθ=A+CBtanθ=BA+C
Standard diagonalization of quadratic form in 2D uses rotation. Choosing θθ such that tan2θ=BA−Ctan2θ=A−CB eliminates the cross term BxyBxy, simplifying classification and standard form conversion.
For pair of straight lines represented by Ax2+2Hxy+By2=0Ax2+2Hxy+By2=0, slopes mm satisfy
A Am2+2Hm+B=0Am2+2Hm+B=0
B Am2−2Hm+B=0Am2−2Hm+B=0
C Bm2+2Hm+A=0Bm2+2Hm+A=0
D Bm2−2Hm+A=0Bm2−2Hm+A=0
Put y=mxy=mx in Ax2+2Hxy+By2=0Ax2+2Hxy+By2=0: x2(A+2Hm+Bm2)=0x2(A+2Hm+Bm2)=0. For nonzero xx, slopes satisfy Bm2+2Hm+A=0Bm2+2Hm+A=0. Roots give slopes of the two lines.
For conic S=0S=0, the polar of point PP and pole of line are related by
A Reciprocity theorem
B Pythagoras theorem
C Sine rule
D Midpoint rule
Pole-polar relation is reciprocal: if point PP lies on polar of QQ, then QQ lies on polar of PP. This is called reciprocity. It’s a key structural property of conics in coordinate geometry.
Maximum number of common tangents for two circles is
A 2
B 3
C 5
D 4
Two separate circles (non-overlapping, not one inside another) can have 4 common tangents: 2 direct and 2 transverse. Tangency or intersection reduces the count. This is a standard circle tangent-count result.
The combined equation of pair of tangents from PP to a circle can be obtained using
A T2=SS1T2=SS1
B T=S1T=S1
C S=S1S=S1
D T=0T=0
For any conic S=0S=0, pair of tangents from point PP is given by T2=SS1T2=SS1. For circle, this yields a second-degree equation representing both tangents together, useful when finding tangency points.
The locus PF/PD=ePF/PD=e gives a hyperbola when
A e=1e=1
B e<1e<1
C e>1e>1
D e=0e=0
Focus-directrix definition classifies conics by eccentricity: ellipse e<1e<1, parabola e=1e=1, hyperbola e>1e>1. When e>1e>1, the set of points has two branches, giving a hyperbola.
For circle x2+y2=r2x2+y2=r2, pair of tangents from point on director circle are
A Parallel
B Perpendicular
C Coincident
D Undefined
Director circle is locus of points from which tangents to the circle are at right angles. So if a point lies on director circle, the two tangents drawn from it meet at 90∘90∘. This follows directly from definition.