Chapter 7: Conic Sections and Their Geometry (Set-4)
Circle with center (2,−1)(2,−1) passing through (5,3)(5,3) has equation
A (x−2)2+(y+1)2=16(x−2)2+(y+1)2=16
B (x+2)2+(y−1)2=16(x+2)2+(y−1)2=16
C (x+2)2+(y−1)2=25(x+2)2+(y−1)2=25
D (x−2)2+(y+1)2=25(x−2)2+(y+1)2=25
With center (2,−1)(2,−1), standard form is (x−2)2+(y+1)2=r2(x−2)2+(y+1)2=r2. Using point (5,3)(5,3): r2=(3)2+(4)2=25r2=(3)2+(4)2=25. So the correct equation is (x−2)2+(y+1)2=25(x−2)2+(y+1)2=25.
For circle x2+y2+2x−4y−4=0x2+y2+2x−4y−4=0, the center is
A (1,−2)(1,−2)
B (2,−1)(2,−1)
C (−1,2)(−1,2)
D (−2,1)(−2,1)
Compare with x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0. Here g=1,f=−2g=1,f=−2. Center is (−g,−f)=(−1,2)(−g,−f)=(−1,2). Completing squares gives (x+1)2+(y−2)2=9(x+1)2+(y−2)2=9.
Distance between centers of x2+y2=25×2+y2=25 and (x−3)2+(y−4)2=16(x−3)2+(y−4)2=16 is
A 33
B 55
C 44
D 77
First center is (0,0)(0,0), second is (3,4)(3,4). Distance is 32+42=532+42=5. This value is useful for deciding intersection/tangency between the circles.
Two circles x2+y2=25×2+y2=25 and (x−3)2+(y−4)2=16(x−3)2+(y−4)2=16 are
A Separate circles
B Touch internally
C Touch externally
D Intersect at two
Radii are 55 and 44; center distance d=5d=5. Since ∣5−4∣=1
Tangent length from P(8,0)P(8,0) to x2+y2=16×2+y2=16 is
A 4848
B 44
C 6060
D 88
Radius r=4r=4, OP=8OP=8. Tangent length =OP2−r2=64−16=48=43=OP2−r2=64−16=48=43. This applies since PP is outside the circle.
If line 3x+4y=123x+4y=12 is tangent to x2+y2=r2x2+y2=r2, then rr equals
A 5/125/12
B 12/512/5
C 33
D 44
For tangency to circle centered at origin, radius equals distance from origin to line. Distance is ∣12∣32+42=12532+42∣12∣=512. So r=12/5r=12/5.
Chord of contact from P(x1,y1)P(x1,y1) to circle x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0 is
A S=0S=0 form
B S1−S2=0S1−S2=0
C S1+S2=0S1+S2=0
D T=0T=0 form
For circle/conic, the chord of contact (polar) from point (x1,y1)(x1,y1) is given by replacing squares using bilinear terms to form T=0T=0. It produces the line joining tangency points.
Director circle of x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0 is
A x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0
B x2+y2=g2+f2−cx2+y2=g2+f2−c
C x2+y2+2gx+2fy+g2+f2=0x2+y2+2gx+2fy+g2+f2=0
D x2+y2=g2+f2+cx2+y2=g2+f2+c
Director circle is locus of points from which tangents to the circle are perpendicular. For circle in general form, it becomes (x+g)2+(y+f)2=0(x+g)2+(y+f)2=0 in shifted form, i.e. x2+y2+2gx+2fy+g2+f2=0x2+y2+2gx+2fy+g2+f2=0.
If tangents from PP to a circle are perpendicular, then PP lies on
A The circle
B Director circle
C Radical axis
D Diameter line
By definition, the director circle is the locus of points from which the pair of tangents to the given circle are at right angles. So perpendicular tangents condition directly means PP lies on director circle.
Radical axis of circles x2+y2=25×2+y2=25 and x2+y2−6x−8y=0x2+y2−6x−8y=0 is
A 3x+4y=253x+4y=25
B 3x+4y=03x+4y=0
C 6x+8y=06x+8y=0
D 6x+8y=256x+8y=25
Write as S1=x2+y2−25=0S1=x2+y2−25=0, S2=x2+y2−6x−8y=0S2=x2+y2−6x−8y=0. Radical axis is S1−S2=0⇒−25+6x+8y=0S1−S2=0⇒−25+6x+8y=0, i.e. 6x+8y=256x+8y=25.
For parabola y2=12xy2=12x, focus is
A (3,0)(3,0)
B (0,3)(0,3)
C (6,0)(6,0)
D (0,6)(0,6)
Compare y2=12xy2=12x with y2=4axy2=4ax. Then 4a=12⇒a=34a=12⇒a=3. Focus for y2=4axy2=4ax is (a,0)(a,0), hence (3,0)(3,0).
Equation of tangent to y2=12xy2=12x at parameter tt is
A ty=x−3t2ty=x−3t2
B y=tx+3ty=tx+3t
C ty=x+3t2ty=x+3t2
D y=2txy=2tx
For y2=4axy2=4ax, tangent at parameter tt is ty=x+at2ty=x+at2. Here a=3a=3. So tangent is ty=x+3t2ty=x+3t2. It touches at (3t2,6t)(3t2,6t).
The normal to y2=4axy2=4ax at (at2,2at)(at2,2at) cuts xx-axis at
A (at2,0)(at2,0)
B (2at2,0)(2at2,0)
C (4at2,0)(4at2,0)
D (0,2at)(0,2at)
A known standard property: for y2=4axy2=4ax, the normal at parametric point (at2,2at)(at2,2at) meets the axis y=0y=0 at (2at2,0)(2at2,0). This comes from the normal equation and intercept calculation.
Focal chord of y2=4axy2=4ax with parameter tt has equation
A ty=x+at2ty=x+at2
B y=−t(x−a)y=−t(x−a)
C y=t(x−a)y=t(x−a)
D y=t(x+a)y=t(x+a)
A focal chord passes through focus (a,0)(a,0). The chord with slope tt through focus has equation y=t(x−a)y=t(x−a) or y=t(x+a)y=t(x+a)? For y2=4axy2=4ax, standard focal chord is y=t(x+a)y=t(x+a). It intersects parabola in a chord through focus.
For parabola x2=8yx2=8y, latus rectum endpoints are
A (±4,2)(±4,2)
B (±2,4)(±2,4)
C (±4,4)(±4,4)
D (±2,2)(±2,2)
Compare x2=8yx2=8y with x2=4ayx2=4ay gives a=2a=2. Focus is (0,2)(0,2). Latus rectum is y=a=2y=a=2. Put y=2y=2: x2=16⇒x=±4×2=16⇒x=±4. Endpoints (±4,2)(±4,2).
Reflection property in parabola means the tangent makes equal angles with
A Axis and chord
B Focal radius and axis
C Directrix and axis
D Latus and axis
At point on parabola, the tangent is the angle bisector between the focal line (line from point to focus) and the line parallel to axis. This explains parallel rays focusing at the focus.
If parabola is translated: (y−2)2=8(x+1)(y−2)2=8(x+1), vertex is
A (1,−2)(1,−2)
B (2,−1)(2,−1)
C (−1,2)(−1,2)
D (−2,1)(−2,1)
Standard right-opening form is (y−k)2=4a(x−h)(y−k)2=4a(x−h). Here (y−2)2=8(x+1)=8(x−(−1))(y−2)2=8(x+1)=8(x−(−1)). So h=−1,k=2h=−1,k=2. Vertex is (−1,2)(−1,2).
For y2=4axy2=4ax, equation of chord with midpoint (x1,y1)(x1,y1) is
A T=ST=S form
B T=S1T=S1 form
C T=S1T=S1 with sign
D T=S1T=S1 always
For any conic S=0S=0, chord with midpoint (x1,y1)(x1,y1) is T=S1T=S1, where TT is bilinear form at (x1,y1)(x1,y1) and S1S1 is value of SS at that point. This is a key chord formula.
For parabola, the point where normal passes through focus is called
A Vertex point
B Focal point
C End of latus
D Focal normal point
A “focal normal” is a normal that passes through the focus. Points on parabola where the normal passes through focus are special and used in chord/normal properties. This is a standard named idea in parabola theory.
Parabola as locus is set of points where distance to focus equals distance to
A Center line
B Axis line
C Directrix line
D Tangent line
Parabola is defined as points PP such that PF=PDPF=PD, where FF is focus and DD is perpendicular distance to directrix. This gives eccentricity e=1e=1 and produces the standard parabola equations.
Ellipse x236+y220=136×2+20y2=1 has foci at
A (±16,0)(±16,0)
B (±4,0)(±4,0)
C (±16,0)(±16,0)
D (±16,0)(±16,0)
Here a2=36⇒a=6a2=36⇒a=6, b2=20b2=20. Then c2=a2−b2=16⇒c=4c2=a2−b2=16⇒c=4. Since major axis is along xx, foci are (±c,0)=(±4,0)(±c,0)=(±4,0).
For ellipse, distance between directrices is
A 2ae2ae
B 2b/e2b/e
C 2be2be
D 2a/e2a/e
For ellipse with major axis along xx, directrices are x=±a/ex=±a/e. Therefore separation between them is 2a/e2a/e. This comes from focus-directrix definition and e=c/ae=c/a.
Normal at (acost,bsint)(acost,bsint) for ellipse has slope
A bcostasintasintbcost
B −bcostasint−asintbcost
C asintbcostbcostasint
D −asintbcost−bcostasint
Tangent slope at (acost,bsint)(acost,bsint) is −bcostasint−asintbcost. Normal slope is negative reciprocal: asintbcostbcostasint. This uses perpendicular slopes rule.
If ellipse has e=35e=53 and a=10a=10, then cc is
A 66
B 55
C 33
D 88
Eccentricity e=c/ae=c/a. So c=ae=10⋅35=6c=ae=10⋅53=6. Then b2=a2−c2=100−36=64⇒b=8b2=a2−c2=100−36=64⇒b=8. This checks ellipse consistency.
The chord of ellipse subtending right angle at center has midpoint lying on
A Circle radius aa
B Circle radius bb
C Auxiliary circle
D Director circle
For ellipse, director circle is locus of points from which tangents are perpendicular; similarly, chords subtending right angle at center relate to director circle concept. In standard results, certain right-angle conditions connect to director circle.
Equation of tangent to ellipse at (x1,y1)(x1,y1) is
A xx1a+yy1b=1axx1+byy1=1
B xx1a2+yy1b2=1a2xx1+b2yy1=1
C xx1+yy1=abxx1+yy1=ab
D x+x1+y+y1=1x+x1+y+y1=1
Replace squares by products in standard ellipse equation. If (x1,y1)(x1,y1) lies on ellipse, the tangent is xx1a2+yy1b2=1a2xx1+b2yy1=1. It meets ellipse at only one point.
For ellipse, sum of distances to foci equals
A 2b2b
B 2c2c
C 2a2a
D a+ba+b
Ellipse is locus of points where PF1+PF2PF1+PF2 is constant. That constant equals the major axis length 2a2a. This remains true for all ellipse points and helps in geometric constructions.
If ellipse major axis along yy, its foci are
A (0,±c)(0,±c)
B (±c,0)(±c,0)
C (±a,0)(±a,0)
D (0,±b)(0,±b)
When major axis is along yy-axis, foci lie on yy-axis. Their distance from center is cc. Thus foci are (0,±c)(0,±c), where c2=a2−b2c2=a2−b2 with aa as semi-major.
For ellipse, conjugate diameter property means each diameter bisects chords parallel to
A Same diameter
B Major axis
C Minor axis
D Other diameter
Two diameters are conjugate if each bisects chords parallel to the other. This is equivalent to a deep symmetry property of ellipse and is used in tangent and chord direction relationships.
The latus rectum line for ellipse (major along xx) is
A y=cy=c
B x=cx=c
C x=ax=a
D y=by=b
Latus rectum passes through focus (c,0)(c,0) and is perpendicular to major axis, hence vertical line x=cx=c. It cuts the ellipse at endpoints of latus rectum, giving length 2b2/a2b2/a.
Hyperbola x29−y216=19×2−16y2=1 has eccentricity
A 4/34/3
B 3/53/5
C 5/45/4
D 5/35/3
Here a2=9⇒a=3a2=9⇒a=3, b2=16⇒b=4b2=16⇒b=4. Then c2=a2+b2=25⇒c=5c2=a2+b2=25⇒c=5. So e=c/a=5/3e=c/a=5/3, which is greater than 1.
Asymptotes of x29−y216=19×2−16y2=1 are
A y=±34xy=±43x
B y=±916xy=±169x
C y=±43xy=±34x
D y=±169xy=±916x
For x2a2−y2b2=1a2x2−b2y2=1, asymptotes are y=±baxy=±abx. Here b/a=4/3b/a=4/3. These lines guide the branches as ∣x∣∣x∣ becomes large.
Hyperbola directrices for standard form are
A x=±a/ex=±a/e
B x=±aex=±ae
C y=±a/ey=±a/e
D y=±aey=±ae
For hyperbola with transverse axis on xx, directrices are vertical lines x=±a/ex=±a/e. Since e>1e>1, these lie closer to origin than foci at ±ae±ae. This is from focus-directrix definition.
Tangent at parametric point (asect,btant)(asect,btant) on hyperbola is
A xcosta−ysintb=1axcost−bysint=1
B xsecta−ytantb=1axsect−bytant=1
C xsecta+ytantb=1axsect+bytant=1
D xsect−ytant=abxsect−ytant=ab
Tangent at (x1,y1)(x1,y1) is xx1a2−yy1b2=1a2xx1−b2yy1=1. Substitute x1=asectx1=asect, y1=btanty1=btant to get xsecta−ytantb=1axsect−bytant=1.
Conjugate hyperbola of x2a2−y2b2=1a2x2−b2y2=1 opens along
A xx-axis
B Both axes
C No axis
D yy-axis
Conjugate hyperbola is x2a2−y2b2=−1a2x2−b2y2=−1, equivalently y2b2−x2a2=1b2y2−a2x2=1. Its positive term is in y2y2, so transverse axis is along yy-axis.
For rectangular hyperbola xy=c2xy=c2, asymptotes are
A y=x,y=−xy=x,y=−x
B x=c,y=cx=c,y=c
C x=0,y=0x=0,y=0
D x+y=0x+y=0
In the form xy=c2xy=c2, as ∣x∣→∞∣x∣→∞, y→0y→0, and as ∣y∣→∞∣y∣→∞, x→0x→0. Hence coordinate axes x=0x=0 and y=0y=0 are asymptotes. This is a key property.
Director circle of x2a2−y2b2=1a2x2−b2y2=1 is
A x2+y2=a2−b2x2+y2=a2−b2
B x2+y2=a2+b2x2+y2=a2+b2
C x2+y2=2a2x2+y2=2a2
D x2+y2=2b2x2+y2=2b2
For standard hyperbola, director circle exists and has equation x2+y2=a2+b2x2+y2=a2+b2. It represents locus of points from which the pair of tangents to hyperbola are perpendicular.
For hyperbola, if point satisfies PF/PD=ePF/PD=e, then ee is
A Equal to 1
B Less than 1
C Equal to 0
D Greater than 1
In focus-directrix definition, conic type depends on eccentricity ee. For hyperbola, the ratio PF/PDPF/PD is a constant greater than 1. This creates an open curve with two branches.
Latus rectum endpoints for x2a2−y2b2=1a2x2−b2y2=1 have x=x=
A aa
B cc
C bb
D 00
Latus rectum passes through focus (c,0)(c,0) and is perpendicular to transverse axis, so it is line x=cx=c. Substituting x=cx=c gives y2=b4a2y2=a2b4, so endpoints are (c,±b2/a)(c,±b2/a).
Transverse axis endpoints of y2a2−x2b2=1a2y2−b2x2=1 are
A (±a,0)(±a,0)
B (0,±b)(0,±b)
C (0,±a)(0,±a)
D (±b,0)(±b,0)
Here transverse axis is along yy-axis. Setting x=0x=0 gives y2=a2y2=a2, so vertices are (0,±a)(0,±a). This helps sketch the hyperbola and locate its branches.
Cone equation x2+y2=4z2x2+y2=4z2 has semi-vertical angle αα with
A tanα=1/2tanα=1/2
B sinα=2sinα=2
C cosα=2cosα=2
D tanα=2tanα=2
Compare x2+y2=z2tan2αx2+y2=z2tan2α with given x2+y2=4z2x2+y2=4z2. Then tan2α=4⇒tanα=2tan2α=4⇒tanα=2 (angle between generator and axis).
A cone’s generator is the line segment from
A Vertex to surface
B Center to focus
C Focus to directrix
D Center to chord
A generator is a straight line on the cone surface passing through the vertex and any point on the base circle. Moving the generator around the axis forms the cone surface. It makes constant semi-vertical angle with axis.
Cone through origin condition in 3D second-degree equation means
A No constant term
B No constant term
C No cross terms
D Only linear terms
For a cone with vertex at origin, the equation is homogeneous of degree 2, so it cannot include constant or linear terms. Presence of constant term would shift surface away from origin, so it wouldn’t pass through vertex at origin.
Intersection of cone with plane parallel to base produces
A Ellipse section
B Parabola section
C Circle section
D Hyperbola section
If plane is parallel to base of a right circular cone, it cuts the cone in a circle. Radius depends on the height at which plane cuts. This is a key link between cones and circular cross-sections.
For general conic, if B2−4AC=0B2−4AC=0, the conic family is
A Ellipse family
B Hyperbola family
C Circle family
D Parabola family
Discriminant B2−4ACB2−4AC classifies second-degree curves. Zero indicates parabola-type (or degenerate case). Negative indicates ellipse/circle-type, and positive indicates hyperbola-type or intersecting lines.
For circle, pole-polar concept: polar of an external point passes through
A Circle center
B Tangency points
C Opposite focus
D Any chord midpoint
The polar (chord of contact) of an external point is the line joining the two tangency points of tangents drawn from that point. This is a core geometric meaning of polar for circles.
For conic S=0S=0, equation T=0T=0 represents
A Tangent always
B Director circle
C Polar line
D Asymptote line
In standard conic notation, T=0T=0 is the polar of point (x1,y1)(x1,y1) with respect to S=0S=0. If the point lies on the conic, T=0T=0 becomes the tangent at that point.
In conic intersections, common chord of two circles is the
A Radical axis
B Diameter line
C Directrix line
D Focus line
When two circles intersect, the chord joining their intersection points is the common chord. This line is exactly the radical axis, because at intersection points powers are equal and S1−S2=0S1−S2=0 holds.
A combined conic locus problem often reduces to eliminating parameter by
A Differentiation
B Substitution
C Integration
D Rotation only
Many locus problems define a moving point using a parameter (like tt). To obtain the locus equation, you substitute parametric relations and eliminate the parameter by algebra, giving a relation in x,yx,y only.
For general conic, translation removes linear terms when new origin is at
A Focus point
B Directrix point
C Center or vertex
D Any intersection
Shifting axes to the center (ellipse/hyperbola/circle) or vertex (parabola) simplifies the equation by reducing or removing linear terms. Then rotation may remove xyxy term, giving standard form clearly.