Chapter 7: Conic Sections and Their Geometry (Set-2)
A circle with diameter endpoints A(x1,y1)A(x1,y1), B(x2,y2)B(x2,y2) satisfies
A PA⋅PB=0PA⋅PB=0
B PA→⊥PB→PA⊥PB
C PA=PBPA=PB
D PA+PB=ABPA+PB=AB
A point P(x,y)P(x,y) lies on the circle with diameter ABAB iff angle APBAPB is 90∘90∘. In vector form, (x−x1,y−y1)⋅(x−x2,y−y2)=0(x−x1,y−y1)⋅(x−x2,y−y2)=0.
For circle x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0, radius squared is
A g2+f2+cg2+f2+c
B c−g−fc−g−f
C g2+f2−cg2+f2−c
D g+f−cg+f−c
Completing squares gives (x+g)2+(y+f)2=g2+f2−c(x+g)2+(y+f)2=g2+f2−c. Hence r2=g2+f2−cr2=g2+f2−c. The circle is real if this value is non-negative.
A tangent to a circle is a line that
A Cuts two points
B Touches one point
C Passes center
D Has fixed slope
A tangent meets a circle at exactly one point (point of contact). In coordinate geometry, this occurs when the line and circle system has one real solution (discriminant zero).
For circle x2+y2=r2x2+y2=r2, slope of radius to (x1,y1)(x1,y1) is
A x1/y1x1/y1
B −y1/x1−y1/x1
C −x1/y1−x1/y1
D y1/x1y1/x1
The radius from origin to (x1,y1)(x1,y1) has slope y1−0x1−0=y1x1x1−0y1−0=x1y1 (when x1≠0x1=0). The tangent slope is the negative reciprocal.
The normal to a circle at a point is the line
A Parallel to tangent
B Perpendicular to radius
C Along the radius
D Bisecting chord
For a circle, the radius at the point of contact is perpendicular to the tangent. Therefore the normal line (perpendicular to tangent) coincides with the radius line passing through the center.
From a point inside a circle, number of real tangents is
A Zero
B One
C Two
D Infinite
From an interior point, any line intersects the circle at two points, so it cannot “just touch” the circle. Tangents exist only from points on the circle (one) or outside the circle (two).
A point PP lies on a circle if its power is
A Positive
B Zero
C Negative
D Maximum
Power of PP w.r.t circle is OP2−r2OP2−r2 (for center OO). If PP is on the circle, OP=rOP=r, so OP2−r2=0OP2−r2=0. This is a quick membership test.
Two circles are orthogonal if at intersection points their tangents are
A Parallel
B Coincident
C Perpendicular
D Equal length
Orthogonal circles cut at right angles. At each intersection point, the tangents are perpendicular (equivalently, the radii to that point are perpendicular). This is a standard geometric condition.
The radical axis of two circles is the locus of points with
A Equal radii
B Equal centers
C Equal chords
D Equal powers
For two circles, points where power w.r.t circle 1 equals power w.r.t circle 2 form a straight line called the radical axis. It is perpendicular to the line joining the centers.
Parametric form of circle x2+y2=r2x2+y2=r2 is
A (rtant,r)(rtant,r)
B (rcost,rsint)(rcost,rsint)
C (rt,r/t)(rt,r/t)
D (r,rcost)(r,rcost)
Taking x=rcostx=rcost, y=rsinty=rsint gives x2+y2=r2(cos2t+sin2t)=r2x2+y2=r2(cos2t+sin2t)=r2. This parametrization traces the full circle as tt varies.
In y2=4axy2=4ax, the vertex is
A (0,0)(0,0)
B (a,0)(a,0)
C (0,a)(0,a)
D (−a,0)(−a,0)
The standard parabola y2=4axy2=4ax is centered at origin in the sense of its vertex location. The vertex is the turning point of the curve, here at (0,0)(0,0).
In y2=4axy2=4ax, the focus is
A (0,a)(0,a)
B (−a,0)(−a,0)
C (a,0)(a,0)
D (0,−a)(0,−a)
For y2=4axy2=4ax, the axis is xx-axis, and the parabola opens right for a>0a>0. The focus lies aa units from the vertex along the axis, so it is (a,0)(a,0).
For y2=4axy2=4ax, the equation of axis is
A x=0x=0
B y=0y=0
C y=xy=x
D y=−xy=−x
The axis of a parabola is its line of symmetry. Since replacing yy by −y−y keeps y2=4axy2=4ax unchanged, symmetry is about the xx-axis, i.e., y=0y=0.
A point (x1,y1)(x1,y1) lies on y2=4axy2=4ax if
A x12=4ay1x12=4ay1
B y1=4ax1y1=4ax1
C x1+y1=4ax1+y1=4a
D y12=4ax1y12=4ax1
Membership means coordinates satisfy the curve’s equation. Substituting (x1,y1)(x1,y1) into y2=4axy2=4ax gives y12=4ax1y12=4ax1. This is the simplest correctness check.
The tangent at parameter tt on y2=4axy2=4ax is
A ty=x−at2ty=x−at2
B y=tx+aty=tx+at
C ty=x+at2ty=x+at2
D y=2txy=2tx
Parametric point is (at2,2at)(at2,2at). The tangent in parameter form is ty=x+at2ty=x+at2. Substituting x=at2,y=2atx=at2,y=2at satisfies it, confirming it touches at that point.
Normal at parameter tt on y2=4axy2=4ax has slope
A tt
B −t−t
C −1/t−1/t
D 1/(2t)1/(2t)
For y2=4axy2=4ax, tangent at parameter tt has slope 1/t1/t. Normal is perpendicular to tangent, so its slope is −t−t. This is a standard basic result for parabola normals.
For x2=4ayx2=4ay, it opens upward when
A a>0a>0
B a<0a<0
C a=0a=0
D Any aa
In x2=4ayx2=4ay, y=x24ay=4ax2. If a>0a>0, yy increases as x2x2 increases, so the parabola opens upward. If a<0a<0, it opens downward.
The latus rectum line for x2=4ayx2=4ay is
A y=−ay=−a
B x=ax=a
C y=ay=a
D x=−ax=−a
Focus is (0,a)(0,a). Latus rectum is the chord through focus perpendicular to axis. Axis is yy-axis, so latus rectum is horizontal through y=ay=a. Endpoints satisfy x2=4a(a)=4a2x2=4a(a)=4a2.
If a parabola has focus (2,0)(2,0) and directrix x=−2x=−2, its equation is
A y2=4xy2=4x
B y2=8xy2=8x
C y2=16xy2=16x
D y2=2xy2=2x
Vertex is midpoint between focus and directrix along axis, so at origin. Distance a=2a=2. Standard form y2=4axy2=4ax gives y2=8xy2=8x. This directly uses focus-directrix definition.
In a parabola, focal chord means a chord that
A Passes vertex
B Parallel to axis
C Perpendicular axis
D Passes focus
Any chord passing through the focus is a focal chord. Latus rectum is a special focal chord perpendicular to axis. The idea helps in chord properties and reflection behavior in basic conic geometry.
In ellipse x2a2+y2b2=1a2x2+b2y2=1, major axis is along
A yy-axis
B y=xy=x
C xx-axis
D y=−xy=−x
When a>ba>b, the larger denominator is under x2x2, meaning the ellipse stretches more in xx-direction. Hence the major axis lies along the xx-axis with endpoints (±a,0)(±a,0).
Foci of ellipse (major along xx-axis) are
A (0,±c)(0,±c)
B (±c,0)(±c,0)
C (±a,0)(±a,0)
D (0,±b)(0,±b)
For ellipse centered at origin with major axis on xx-axis, foci lie on that axis at distance cc from center. The relation c2=a2−b2c2=a2−b2 places them at (±c,0)(±c,0).
Ellipse becomes a circle when
A a=ba=b
B a=0a=0
C b=0b=0
D c=ac=a
If a=b=ra=b=r, ellipse equation becomes x2r2+y2r2=1r2x2+r2y2=1 i.e. x2+y2=r2x2+y2=r2, a circle. Then eccentricity becomes 0 because both foci coincide at center.
The director circle of ellipse exists only when
A e>1e>1
B e=1e=1
C e<1/2e<1/2
D e=0e=0 only
For ellipse, director circle equation involves a2−2b2a2−2b2. It is real only if a2>2b2a2>2b2, which converts to e2<1/2e2<1/2 i.e. e<1/2e<1/2. Otherwise it is imaginary.
Tangent at parametric point (acost,bsint)(acost,bsint) is
A xsinta+ycostb=1axsint+bycost=1
B xcosta+ysintb=1axcost+bysint=1
C axcost+bysint=1axcost+bysint=1
D xcost+ysint=abxcost+ysint=ab
Using xx1a2+yy1b2=1a2xx1+b2yy1=1 with x1=acost,y1=bsintx1=acost,y1=bsint gives xcosta+ysintb=1axcost+bysint=1. It touches ellipse at that parametric point.
For ellipse, distance between foci is
A 2a2a
B 2b2b
C a+ba+b
D 2c2c
Foci are (±c,0)(±c,0) or (0,±c)(0,±c). The separation is twice the focus distance from center, hence 2c2c. Since c2=a2−b2c2=a2−b2, this depends on how stretched the ellipse is.
The focal distance property of ellipse means PF1+PF2PF1+PF2 is
A 2b2b
B 2c2c
C 2a2a
D Variable
By definition, ellipse is locus where sum of distances from two fixed foci is constant. That constant equals major axis length 2a2a. This remains true for every point on ellipse, not just special points.
Minor axis endpoints of ellipse (major along xx-axis) are
A (0,±b)(0,±b)
B (±b,0)(±b,0)
C (±a,0)(±a,0)
D (0,±a)(0,±a)
The minor axis is along yy-axis when major axis is along xx-axis. Setting x=0x=0 in equation gives y2=b2y2=b2, so endpoints are (0,±b)(0,±b). This is basic coordinate reading.
The latus rectum endpoints (major along xx-axis) have x=x=
A 00
B cc
C aa
D bb
Latus rectum passes through focus (c,0)(c,0) and is perpendicular to major axis, so it is vertical line x=cx=c. Substituting x=cx=c into ellipse gives y=±b2ay=±ab2.
Conjugate diameter idea in ellipse refers to
A Equal radii
B Perpendicular tangents
C Parallel tangents set
D Same focus distance
Two diameters are conjugate if each bisects chords parallel to the other. Equivalently, endpoints of one diameter have tangents parallel to the other diameter’s direction. It’s an important geometric concept, even at intro level.
Hyperbola y2a2−x2b2=1a2y2−b2x2=1 opens along
A xx-axis
B Line y=xy=x
C Line y=−xy=−x
D yy-axis
The positive term is y2a2a2y2, so the transverse axis is along yy-axis. Vertices are (0,±a)(0,±a). The branches open upward and downward, approaching asymptotes as ∣y∣∣y∣ increases.
Asymptotes of y2a2−x2b2=1a2y2−b2x2=1 are
A y=±abxy=±bax
B y=±baxy=±abx
C x=±bayx=±aby
D x=±abyx=±bay
Set the hyperbola equation to 0: y2a2−x2b2=0a2y2−b2x2=0. This gives ya=±xbay=±bx, so y=±abxy=±bax. These are guiding lines.
Vertices of x2a2−y2b2=1a2x2−b2y2=1 are
A (0,±a)(0,±a)
B (±c,0)(±c,0)
C (±a,0)(±a,0)
D (0,±c)(0,±c)
Vertices lie where hyperbola crosses its transverse axis. Putting y=0y=0 gives x2=a2x2=a2, so x=±ax=±a. Hence vertices are (±a,0)(±a,0). This is a basic graphing fact.
For hyperbola, foci lie at
A (0,±c)(0,±c)
B (±c,0)(±c,0)
C (±b,0)(±b,0)
D (0,±b)(0,±b)
In standard hyperbola x2a2−y2b2=1a2x2−b2y2=1, foci are on transverse axis, so at (±c,0)(±c,0). Here c2=a2+b2c2=a2+b2, so c>ac>a.
Conjugate axis length of x2a2−y2b2=1a2x2−b2y2=1 is
A 2a2a
B a+ba+b
C 2c2c
D 2b2b
For hyperbola, the conjugate axis corresponds to the negative term’s denominator. Here it is bb. So full conjugate axis length is 2b2b. It helps define the asymptote rectangle used in sketching.
The latus rectum length of a hyperbola is
A 2a2/b2a2/b
B 2ab2ab
C 2b2/a2b2/a
D a2+b2a2+b2
For x2a2−y2b2=1a2x2−b2y2=1, the latus rectum through focus has length 2b2a2ab2. It is a focal chord perpendicular to transverse axis, similar in structure to ellipse formula.
Rectangular hyperbola has eccentricity
A 22
B 22
C 1/21/2
D 11
Rectangular hyperbola has a=ba=b. Then c2=a2+b2=2a2c2=a2+b2=2a2 so c=a2c=a2. Hence e=c/a=2e=c/a=2. Its asymptotes become perpendicular, giving the “rectangular” name.
Conjugate hyperbola of x2a2−y2b2=1a2x2−b2y2=1 is
A x2a2−y2b2=−1a2x2−b2y2=−1
B x2a2+y2b2=1a2x2+b2y2=1
C y2a2−x2b2=1a2y2−b2x2=1
D x2−y2=a2x2−y2=a2
Conjugate hyperbola shares the same asymptotes and center, but the constant changes sign. Replacing 1 by −1−1 gives the conjugate. It opens along the conjugate axis direction, swapping orientation.
A point satisfies a hyperbola if it makes ∣PF1−PF2∣∣PF1−PF2∣ equal to
A 2b2b
B 2c2c
C 2a2a
D a+ba+b
Hyperbola is defined as locus where absolute difference of distances to the two foci is constant. That constant equals transverse axis length 2a2a. This matches the standard form and explains two-branch nature.
Director circle of hyperbola 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=b2x2+y2=b2
For standard hyperbola, director circle is real and is given by x2+y2=a2+b2x2+y2=a2+b2. It is connected with points from which tangents to the hyperbola are perpendicular, an important basic concept.
A cone with vertex at origin has equation that is
A Non-homogeneous linear
B Cubic homogeneous
C Exponential form
D Homogeneous quadratic
A cone through origin is represented by a second-degree homogeneous equation in x,y,zx,y,z, meaning every term has total degree 2. This ensures scaling: if (x,y,z)(x,y,z) lies on it, so does (kx,ky,kz)(kx,ky,kz).
Right circular cone with axis zz-axis and semi-vertical angle αα satisfies
A x2+y2=z2cot2αx2+y2=z2cot2α
B x2+y2=z2tan2αx2+y2=z2tan2α
C x2+y2=ztanαx2+y2=ztanα
D x2+y2=tan2αx2+y2=tan2α
For a cone symmetric about zz-axis, radius at height zz is ∣z∣tanα∣z∣tanα. Squaring gives x2+y2=(ztanα)2=z2tan2αx2+y2=(ztanα)2=z2tan2α. This is the standard cone equation.
A plane cuts a cone perpendicular to its axis; the section is
A Circle
B Parabola
C Ellipse
D Hyperbola
If the cutting plane is perpendicular to the cone’s axis, it intersects the cone in a circle (for one nappe). This is because the intersection at a fixed height has constant distance from the axis, giving circular symmetry.
A plane cutting both nappes of a cone produces a
A Circle
B Parabola
C Hyperbola
D Ellipse
When the cutting plane intersects both nappes (upper and lower parts) of the cone, it produces two separated branches. That is exactly the shape of a hyperbola, matching the definition and standard equation type.
A conic is defined using focus-directrix by ratio
A PF+PD=ePF+PD=e
B PF/PD=ePF/PD=e
C PF−PD=ePF−PD=e
D PF×PD=ePF×PD=e
For any conic, distance from point PP to focus FF divided by perpendicular distance to directrix DD is constant ee (eccentricity). Ellipse: e<1e<1, parabola: e=1e=1, hyperbola: e>1e>1.
In general second-degree equation, BxyBxy term indicates possible
A Translation only
B Always circle
C Always parabola
D Rotation needed
If B≠0B=0 in Ax2+Bxy+Cy2+Dx+Ey+F=0Ax2+Bxy+Cy2+Dx+Ey+F=0, the axes are generally tilted. A rotation of axes can remove the xyxy term, simplifying identification of the conic in standard position.
Condition for tangency of a line to a conic often comes from
A Minimum value only
B Maximum slope
C Discriminant zero
D Equal intercepts
When substituting a line equation into a conic equation, you get a quadratic in one variable. For tangency, the line must meet the conic in exactly one point, so the quadratic has equal roots, meaning discriminant equals zero.
A chord of a conic is a line segment joining
A Two centers
B Two curve points
C Two foci
D Two directrices
A chord is the segment joining two points on the curve. For circle it’s familiar, but same idea holds for ellipse, parabola, and hyperbola. A diameter is a special chord passing through the center (when it exists).
Translation of axes changes the conic equation mainly by
A Changing linear terms
B Removing x2x2 terms
C Changing degree
D Removing constant
Shifting origin (x=X+h, y=Y+k)(x=X+h, y=Y+k) keeps the equation quadratic, but it changes linear and constant terms after expansion. It is used to move a conic to standard position by relocating its center or vertex.
Intersection points of two conics are found by solving
A One equation only
B Only slope condition
C Two equations together
D Only parameter form
Intersection points satisfy both conic equations simultaneously. Solving the system gives common solutions, which can be 0, 1, 2, 3, or 4 (maximum 4 for two second-degree curves). This is basic intersection logic.