Chapter 7: Conic Sections and Their Geometry (Set-1)
For (x−h)2+(y−k)2=r2(x−h)2+(y−k)2=r2, what does (h,k)(h,k) represent
A Focus of circle
B Midpoint of chord
C Center of circle
D Point of tangency
In the standard circle form, shifting xx by hh and yy by kk moves the circle’s location. The point (h,k)(h,k) is the center, and rr is the radius.
In x2+y2+2gx+2fy+c=0x2+y2+2gx+2fy+c=0, the center is
A (g,f)(g,f)
B (f,g)(f,g)
C (−f,−g)(−f,−g)
D (−g,−f)(−g,−f)
Rewrite as (x+g)2+(y+f)2=g2+f2−c(x+g)2+(y+f)2=g2+f2−c. The squared terms show the center at (−g,−f)(−g,−f). This is the basic way to identify center in general form.
A circle is real if its radius squared is
A Positive or zero
B Negative number
C Zero always
D Always positive
A real circle requires r2≥0r2≥0. If r2>0r2>0, it’s a proper circle. If r2=0r2=0, it becomes a point circle. If r2<0r2<0, no real points exist.
The length of a tangent from point PP to circle is
A OP−rOP−r
B OP2−r2OP2−r2
C OP−rOP−r
D OP2−r2OP2−r2
If OO is the center and OPOP is distance from center to external point, then in right triangle OPTOPT, OT=rOT=r. So PT=OP2−r2PT=OP2−r2 by Pythagoras.
Power of point PP w.r.t circle (center OO, radius rr) is
A OP2+r2OP2+r2
B OP−r2OP−r2
C r2−OPr2−OP
D OP2−r2OP2−r2
Power of a point measures how far the point is from the circle in squared-distance sense. It equals OP2−r2OP2−r2. It is positive outside, zero on the circle, and negative inside.
Tangent at point (x1,y1)(x1,y1) on x2+y2=r2x2+y2=r2 is
A xx1+yy1=r2xx1+yy1=r2
B xx1+yy1=0xx1+yy1=0
C x+x1+y+y1=rx+x1+y+y1=r
D x2+y2=r2x2+y2=r2
For a circle centered at origin, the tangent at (x1,y1)(x1,y1) is obtained by replacing squares with products: xx1+yy1=r2xx1+yy1=r2. It touches the circle at exactly one point.
The tangent is perpendicular to the radius at the point of contact because
A Tangent passes center
B Radius is zero
C Circle has no axes
D Radius normal line
At the touching point, the radius joins center to contact point. The tangent just “grazes” the circle, so it must be perpendicular to that radius. Hence the radius acts as the normal at contact.
Condition that line ax+by+c=0ax+by+c=0 touches x2+y2=r2x2+y2=r2 is
A c=r(a+b)c=r(a+b)
B c2=r(a2+b2)c2=r(a2+b2)
C c2=r2(a2+b2)c2=r2(a2+b2)
D c=r2(a+b)c=r2(a+b)
Distance from origin to line is ∣c∣a2+b2a2+b2∣c∣. For tangency, distance equals radius rr. Squaring gives c2=r2(a2+b2)c2=r2(a2+b2), a standard tangency test.
A circle through three non-collinear points is
A Unique circle
B Impossible always
C Not unique
D Infinite circles
Three non-collinear points determine a unique circle because they fix center as intersection of perpendicular bisectors of two chords. Collinear points cannot form a circle uniquely since no finite radius fits.
Two circles intersect in how many points at most
A One point
B Three points
C Four points
D Two points
Two circles can be separate (0), tangent (1), intersecting (2), or coincident (infinite). Geometrically, circle-circle intersection gives at most two points because solving two equations yields up to two solutions.
Standard form of a parabola opening right is
A x2=4ayx2=4ay
B y2=4axy2=4ax
C y2=−4axy2=−4ax
D x2=−4ayx2=−4ay
For y2=4axy2=4ax, the axis is along positive xx-direction, vertex at origin, and focus at (a,0)(a,0). The sign of aa decides opening direction; for a>0a>0, it opens right.
For x2=4ayx2=4ay, the focus is
A (0,a)(0,a)
B (a,0)(a,0)
C (−a,0)(−a,0)
D (0,−a)(0,−a)
In x2=4ayx2=4ay, the axis is the yy-axis. Vertex is at origin and the focus lies aa units above the vertex. So focus is (0,a)(0,a), and directrix is y=−ay=−a.
Eccentricity of a parabola equals
A e=0e=0
B e<1e<1
C e>1e>1
D e=1e=1
A parabola is defined as locus where distance from focus equals distance from directrix. That condition gives eccentricity exactly 1. This is a key identity distinguishing parabola from ellipse (e<1)(e<1) and hyperbola (e>1)(e>1).
Length of latus rectum of y2=4axy2=4ax is
A 2a2a
B aa
C 4a4a
D 8a8a
Latus rectum is focal chord perpendicular to axis passing through focus. For y2=4axy2=4ax, it lies at x=ax=a. Substituting x=ax=a gives y2=4a2y2=4a2 so endpoints are y=±2ay=±2a, length 4a4a.
Parametric point on y2=4axy2=4ax can be written as
A (at2,2at)(at2,2at)
B (2at,at2)(2at,at2)
C (a+t,2at)(a+t,2at)
D (t,at)(t,at)
Using parameter tt, we set y=2aty=2at and x=at2x=at2. Then y2=4a2t2=4a(at2)=4axy2=4a2t2=4a(at2)=4ax, so it satisfies the parabola equation. This is a standard basic parametrization.
Tangent to y2=4axy2=4ax at (x1,y1)(x1,y1) is
A yy1=a(x+x1)yy1=a(x+x1)
B yy1=2a(x+x1)yy1=2a(x+x1)
C yy1=4a(x+x1)yy1=4a(x+x1)
D y+y1=2a(x+x1)y+y1=2a(x+x1)
For y2=4axy2=4ax, the tangent at (x1,y1)(x1,y1) is obtained by replacing y2y2 with yy1yy1 and xx with x+x122x+x1 logic, giving yy1=2a(x+x1)yy1=2a(x+x1).
The directrix of y2=4axy2=4ax is
A x=ax=a
B y=ay=a
C y=−ay=−a
D x=−ax=−a
In y2=4axy2=4ax, focus is (a,0)(a,0). Directrix lies opposite to focus at distance aa from vertex along axis, so it is the vertical line x=−ax=−a. This matches the focus-directrix definition.
The axis of x2=4ayx2=4ay is along
A yy-axis
B xx-axis
C Line y=xy=x
D Line y=−xy=−x
The axis is the line about which the parabola is symmetric. In x2=4ayx2=4ay, replacing xx with −x−x keeps equation same, so symmetry is about the yy-axis, making it the parabola’s axis.
If vertex is (h,k)(h,k) and axis parallel to xx-axis, basic parabola form is
A (x−h)2=4a(y−k)(x−h)2=4a(y−k)
B (y−h)2=4a(x−k)(y−h)2=4a(x−k)
C (x−k)2=4a(y−h)(x−k)2=4a(y−h)
D (y−k)2=4a(x−h)(y−k)2=4a(x−h)
Axis parallel to xx-axis means the parabola opens right/left. So squared term is in yy. Shifting vertex to (h,k)(h,k) gives (y−k)2=4a(x−h)(y−k)2=4a(x−h), the translated standard form.
Reflection property of a parabola says
A Radius equals focus
B Focus rays reflect parallel
C Tangent equals chord
D Directrix is midpoint
A ray from focus to a point on the parabola reflects off the tangent and travels parallel to the axis (and vice versa). This is why parabolic reflectors are used in headlights and satellite dishes.
Standard ellipse centered at origin with major axis on xx-axis is
A x2a2+y2b2=1a2x2+b2y2=1
B x2b2+y2a2=1b2x2+a2y2=1
C xa+yb=1ax+by=1
D x2+y2=a2x2+y2=a2
For an ellipse with major axis along xx-axis, a>ba>b. The standard form becomes x2a2+y2b2=1a2x2+b2y2=1. Foci lie on xx-axis at (±c,0)(±c,0).
For ellipse x2a2+y2b2=1a2x2+b2y2=1, relation among a,b,ca,b,c is
A c2=a2+b2c2=a2+b2
B c2=b2−a2c2=b2−a2
C c=a+bc=a+b
D c2=a2−b2c2=a2−b2
In an ellipse, distance from center to focus is cc, and it satisfies c2=a2−b2c2=a2−b2 where aa is semi-major and bb is semi-minor. This ensures c
Eccentricity of an ellipse is
A e>1e>1
B e=1e=1
C 0
D e=0e=0 only
Ellipse is “less stretched” than a parabola. Its eccentricity is e=cae=ac where c
Length of latus rectum of an ellipse is
A 2b2aa2b2
B 2a2bb2a2
C b2aab2
D 2ab2ab
Latus rectum is focal chord perpendicular to major axis through focus. Its full length for ellipse is 2b2a2ab2. It becomes shorter as ellipse gets more stretched (larger aa compared to bb).
Parametric point on ellipse x2a2+y2b2=1a2x2+b2y2=1 is
A (asint,bcost)(asint,bcost)
B (atant,b)(atant,b)
C (at,bt)(at,bt)
D (acost,bsint)(acost,bsint)
Substituting x=acostx=acost and y=bsinty=bsint gives cos2t+sin2t=1cos2t+sin2t=1. So the point lies on ellipse for any real tt. This is the most common parametrization.
Tangent to ellipse at (x1,y1)(x1,y1) is
A xx1a+yy1b=1axx1+byy1=1
B xx1a2+yy1b2=1a2xx1+b2yy1=1
C xx1+yy1=1x1x+y1y=1
D xx1+yy1=a2xx1+yy1=a2
For ellipse, tangent at a point is formed by replacing x2x2 with xx1xx1 and y2y2 with yy1yy1 in the standard equation. This line touches ellipse at exactly one point.
The auxiliary circle of ellipse x2a2+y2b2=1a2x2+b2y2=1 is
A x2+y2=b2x2+y2=b2
B x2+y2=c2x2+y2=c2
C x2+y2=a2x2+y2=a2
D x2+y2=abx2+y2=ab
Auxiliary circle uses semi-major axis as radius. For ellipse with semi-major aa, the auxiliary circle is x2+y2=a2x2+y2=a2. It helps relate ellipse points with angles through parametric form and projections.
For ellipse, sum of distances from a point to two foci is
A Constant 2b2b
B Constant 2c2c
C Not constant
D Constant 2a2a
Ellipse is locus where sum of distances to two fixed foci is constant. That constant equals length of major axis, 2a2a. This property defines an ellipse even without using coordinate equations.
Area of ellipse with semi-axes aa and bb is
A πabπab
B 2πab2πab
C π(a+b)π(a+b)
D πa2πa2
Ellipse area scales like circle area. A circle radius rr has area πr2πr2. Stretching a circle by factor a/ra/r in xx and b/rb/r in yy gives area πabπab.
The major axis length of ellipse is
A aa
B bb
C 2a2a
D 2b2b
In standard ellipse, aa is semi-major axis (half of major axis). So full major axis length is 2a2a. Similarly, minor axis length is 2b2b. This is basic but very frequently used.
Standard hyperbola with transverse axis along xx-axis is
A x2a2+y2b2=1a2x2+b2y2=1
B y2a2−x2b2=1a2y2−b2x2=1
C x2−y2=a2x2−y2=a2
D x2a2−y2b2=1a2x2−b2y2=1
Hyperbola differs from ellipse by a minus sign. With transverse axis on xx-axis, the standard form is x2a2−y2b2=1a2x2−b2y2=1. It has two separate branches opening left and right.
For hyperbola x2a2−y2b2=1a2x2−b2y2=1, relation is
A c2=a2+b2c2=a2+b2
B c2=a2−b2c2=a2−b2
C c2=b2−a2c2=b2−a2
D c=a−bc=a−b
For hyperbola, focal distance cc is greater than aa. It satisfies c2=a2+b2c2=a2+b2. This makes eccentricity e=ca>1e=ac>1, a key identity separating hyperbola from ellipse.
Eccentricity of a hyperbola satisfies
A e=1e=1
B e>1e>1
C e<1e<1
D e=0e=0
Hyperbola is more “open” than parabola. Since c>ac>a, e=cae=ac becomes greater than 1. This eccentricity tells how far the foci are compared to the transverse semi-axis.
Asymptotes of x2a2−y2b2=1a2x2−b2y2=1 are
A y=±abxy=±bax
B y=±abxy=±abx
C x=±yx=±y
D y=±baxy=±abx
Asymptotes come from setting the hyperbola equation equal to zero: x2a2−y2b2=0a2x2−b2y2=0. This simplifies to y=±baxy=±abx. Branches approach these lines far away.
Parametric point on x2a2−y2b2=1a2x2−b2y2=1 is
A (asect,btant)(asect,btant)
B (acost,bsint)(acost,bsint)
C (atant,bsect)(atant,bsect)
D (at,bt)(at,bt)
Using identity sec2t−tan2t=1sec2t−tan2t=1, set x=asectx=asect and y=btanty=btant. Substituting gives sec2t1−tan2t1=11sec2t−1tan2t=1, satisfying the hyperbola equation.
Tangent to hyperbola at (x1,y1)(x1,y1) is
A xx1a2+yy1b2=1a2xx1+b2yy1=1
B xx1−yy1=a2xx1−yy1=a2
C xx1a2−yy1b2=1a2xx1−b2yy1=1
D x+x1=y+y1x+x1=y+y1
Similar to ellipse, tangent is formed by replacing squares with products, but keeping the minus sign. So the tangent becomes xx1a2−yy1b2=1a2xx1−b2yy1=1. This line touches the hyperbola at one point.
Rectangular hyperbola has asymptotes that are
A Parallel lines
B Coincident lines
C No asymptotes
D Perpendicular lines
For a rectangular hyperbola, the asymptotes are perpendicular. In the standard form x2−y2=a2x2−y2=a2 (or xy=c2xy=c2 after rotation), asymptotes meet at right angles, hence “rectangular.”
For hyperbola, difference of distances to two foci is
A Constant 2b2b
B Constant 2a2a
C Constant 2c2c
D Always zero
Hyperbola is locus where absolute difference of distances from two fixed foci is constant. That constant equals length of transverse axis, 2a2a. This definition matches the standard equation and explains the two-branch shape.
Transverse axis length of x2a2−y2b2=1a2x2−b2y2=1 is
A 2a2a
B aa
C bb
D 2b2b
In hyperbola, aa is semi-transverse axis. So the full transverse axis is 2a2a. It is the segment between the two vertices (±a,0)(±a,0). This is a common basic property used in graphs.
A hyperbola centered at (h,k)(h,k) with transverse axis parallel xx-axis is
A (x−h)2a2+(y−k)2b2=1a2(x−h)2+b2(y−k)2=1
B (y−k)2a2−(x−h)2b2=1a2(y−k)2−b2(x−h)2=1
C (x−h)2+(y−k)2=r2(x−h)2+(y−k)2=r2
D (x−h)2a2−(y−k)2b2=1a2(x−h)2−b2(y−k)2=1
Translating the center from origin to (h,k)(h,k) replaces xx with (x−h)(x−h) and yy with (y−k)(y−k). With transverse axis along xx-direction, the minus sign stays on yy-term.
A general second-degree equation represents a conic because it is
A First-degree equation
B Always linear pair
C Second-degree equation
D Always a circle
Conic sections in coordinate geometry are represented by quadratic (second-degree) equations in xx and yy. Depending on coefficients, the curve can be circle, parabola, ellipse, hyperbola, or a degenerate case like lines.
In Ax2+Bxy+Cy2+Dx+Ey+F=0Ax2+Bxy+Cy2+Dx+Ey+F=0, if B2−4AC=0B2−4AC=0 the conic is
A Circle type
B Ellipse type
C Hyperbola type
D Parabola type
The discriminant B2−4ACB2−4AC classifies conics. When it equals zero, the conic is a parabola (or a degenerate form). Positive means hyperbola type, negative means ellipse/circle type.
If B2−4AC<0B2−4AC<0, the conic is
A Hyperbola type
B Ellipse type
C Parabola type
D Pair of lines
Negative discriminant indicates ellipse-type conic (including circle as a special case). The curve is closed if real. If additional conditions fail, it may be imaginary or a point, but the type is ellipse-family.
If B2−4AC>0B2−4AC>0, the conic is
A Hyperbola type
B Circle type
C Parabola type
D No conic
Positive discriminant means hyperbola-family. The equation can represent a hyperbola or a pair of intersecting straight lines (degenerate). This test is a basic and fast way to identify conic type from coefficients.
A pair of straight lines can occur when the conic is
A Always non-degenerate
B Always closed curve
C Only circle case
D Degenerate quadratic
When a second-degree equation factors into two linear factors, it represents a pair of straight lines. Such cases are called degenerate conics. They occur under special coefficient relationships, not for a general non-degenerate conic.
The condition for a right circular cone uses a fixed angle between
A Generator and base
B Base and axis
C Axis and generator
D Axis and plane
A right circular cone is formed by lines (generators) through the vertex making a constant semi-vertical angle with the axis. That fixed angle defines the cone’s spread and is crucial in its standard 3D equation forms.
A cone is a quadratic surface because its equation is
A Quadratic in variables
B Linear in variables
C Cubic polynomial
D Trigonometric only
A cone in 3D is represented by a homogeneous second-degree equation in x,y,zx,y,z (for a cone with vertex at origin). The squared terms and cross terms make it a quadratic surface in analytic geometry.
When a plane cuts a cone parallel to a generator, the section is
A Circle section
B Ellipse section
C Hyperbola section
D Parabola section
Conic sections come from cutting a cone with a plane. If the plane is parallel to a generator line, it produces a parabola. This matches the parabola’s eccentricity e=1e=1 and its open, single-branch shape.
Pole and polar idea mainly connects a point with a
A Circle radius
B Tangent line family
C Triangle sides
D Coordinate axes
Pole-polar relates a point to a line (its polar) with respect to a conic. It is built using tangents and chord of contact ideas. Even at basic level, it links external points to tangent constructions.
For a circle, chord of contact from an external point is
A Diameter through point
B Locus of centers
C Tangent perpendicular axis
D Line joining tangency points
From an external point, two tangents can be drawn to a circle (if outside). The chord joining the two points of tangency is called the chord of contact. It represents the “touching” chord associated with that point