Practice Test


1) A normal to any circle always lies along its


2) If lines 12x + 5y + 16 = 0 and 12x + 5y - 10 = 0 both touch the same circle, then radius of this circle is


3) If equation of common tangent at (1, -1) to two circles, each of radius 13, is 12x + 5y - 7 = 0, then centres of circles are


4) Line 3x + y = 0 touches a circle centered at (2,-1). Then, other tangent to this circle from the origin is


5) If two circles of same radius r and centres at (2, 3) and (5, 6) cut each other orthogonally, then r =


6) Radius of the larger circle in the first quadrant which touches the line 4x + 3y - 12 = 0, and the co-ordinate axes, is


7) Centre of the circle lies on the line x + 2y = 3 as well as on the bisector of the angle in the first quadrant. If this circle touches the line 4x + 8y = 7, then its area is


8) A circle touches both the axes, and the line 4x + 3y = 6 in the first quadrant. If it lies entirely below the line, then its equation is


9) The equation of the circle having x - y - 2 = 0 and x - y + 2 = 0 as two tangents, and x - y = 0 as a diameter, is


10) If two circles cut each other orthogonally, then


11) A circle passes through the origin (2, 0) and (0, 4). Co-ordinates of its center are


12) Equation of the circle passing through the origin, and the points of intersection of the line 4x + 3y = 12 with the co-ordinate axes, is


13) If 2x + 3y = 7 and x - 3y = -10 are two diameters of a circle passing through (11, -2), then the length of its diameter is


14) Equation of the circle having 12x + 5y + 7 = 0 as a tangent, and x + y = 1, x - y = 3 as two normals, is


15) Equation of the circum - circle of the tangent formed by the lines x = 0, y = 0 and 3x - 4y = 12 is


16) If y = 2x is a tangent to a circle centred at (1,1), then the other tangent from the origin to this circle is


17) If a circle whose centre is at (1,2) touches the line x+ 3y = 3, then their point of contact is


18) Tangents at the ends of a focal chord of a parabola meet each other


19) The only real normal to a parabola, passing through its focus, is its


20) Point of intersection of two perpendicular tangents to a parabola lies on the


21) Equation of the parabola with vertex at the origin, Y-axis as the axis and passing through (3, 9) is


22) Two lines x - 2y = 4 and 7x + 2y + 20 = 0 touch an ellipse whose equation is


23) If the line 3x + 2y = 8 touches the ellipse whose semi-minor axis is 2, then the equation of this ellipse is


24) If the product of lengths of perpendiculars drawn from the foci of a hyperbola to its tangent 2x - y + 4 = 0 is 9, then the equation of this hyperbola is


25) If conjugate axis equals latus-rectum, then: e =


26) Find the equation of the circle which touches the both axes in first quadrant and whose radius is a.


27) Equation of a circle which passes through (3,6) and touches the axes is


28) Equation of the circle with centre on the y-axis and passing through the origin and the point (2,3) is


29) If the lines 2x - 3y = 5 and 3x - 4y = 7 are the diameters of a circle of area 154 sq units, then find the equation of the circle.


30) Find the equation of the circle which passes through the points (2,3) and (4,5) and the centre lies on the straight line y - 4x + 3 = 0.


31) The normal at the point (3,4) on a circle cuts the circle at the point (-1, -2). Then, the equation of the circle is


32) If the lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to a circle, then find the radius of the circle.


33) Find the equation of a circle which touches both the axes and the line 3x - 4y + 8 = 0 and lies in the third quadrant.


34) Two circles with centres (2,3) and (5,6) cut orthogonally. If radius of both circles are equal, then radius is equal to


35) If two circles, each of radius 5 units, touch each other at (1,2) and the equation of their common tangent is 4x + 3y = 10, then equation of the circle a portion of which lies in all the quadrants, is


36) The area of the circle centred at (1,2) and passing through (4,6) is


37) If a circle passes through the point (0, 0), ( a, 0) and (0, b), then find the coordinates of its centre.


38) The distinct points A ( 0,0 ), B ( 0,1 ), C ( 1,0 ) and D ( 2a, 3a) are concyclic, then


39) The equation of a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is


40) Let PQ and PS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point x on the circumference of the circle, then 2r equals


41) Two rods of lengths a and b slide along the x-axis and y-axis respectively in such a manner that their ends are concyclic. The locus of the centre of the circle passing through the end points is


42) Statement I: Number of circles passing through (-2, 1), (-1, 0), (-4, 3) is 1.
Statement II: Through three non-collinear points in a plane only one circle can be drawn.


43) The circle passing through (1, -2) and touching the axis to x at (3,0) also passes through the point


44) The length of the diameter of the circle which touches the x-axis at the point (1,0) and passes through the point (2,3) is


45) The equation of the circle passing through the points (1,0) and (0,1) and having the smallest radius is


46) Consider a family of circles which are passing through the point (-1,1) and are tangent to x-axis. If (h,k) is the centre of circle, then


47) A circle touches the x-axis and also touches the circle with centre at ( 0,3) and radius 2. The locus of the centre of the circle is


48) The smallest circle with centre on y- axis and passing through the point (7,3) has radius


49) A variable circle passes through the first point A ( p,q) and touches x-axis. The locus of the other end of the diameter through A is


50) The equation of latusrectum of a parabola is x + y = 8 and the equation of the tangent at the vertex is x + y =12, then length of the latusrectum is


51) If the focus of a parabola is ( 0, -3 ) and its directrix is y = 3 and its equation is


52) If the vertex of the parabola is the point ( -3, 0 ) and the directrix is the line x + 5 = 0, then its equation is


53) If the point ( 0, 4 ) and (0, 2) are respectively the vertex and focus of a parabola, then the equation of the parabola is


54) The equation of parabola having vertex ( 0, 0 ) passing through ( 2, 3 ) and axis is along x-axis is


55) Tangent at the vertex divides the distance between directrix and latusrectum in the ratio


56) If the tangles at P and Q on the parabola meet in T, then SP, ST and SQ are in


57) If the tangent and normal at any point P of a parabola meet the axes in T and G respectively, then


58) If a parabola reflector is 20 cm in diameter and 5 cm deep, then the focus is


59) The equation of the parabola whose focus is the point ( 0, 0 ) and the tangent at the vertex is x - y + 1 = 0, is


60) The equation of the parabola whose focus is the point ( 0, 0 ) and the tangent at the vertex is x - y + 1 = 0 is


61) The radius of circle when a attains its maximum value


62) The slope of the tangent when radius of the circle is maximum is


63) The minimum area bounded by the tangent and the coordinate axes is


64) A parabola has the origin as its focus and the line x = 2 as the directrix. Then, the vertex of the parabola is at


65) The locus of a point which moves such that the sum of its distances from two fixed points is always a constant, is


66) Find the equation of an ellipse, if major axis on the x-axis and passes through the points ( 4, 3 ) and ( 6, 2 ).


67) The sum of focal distance of any point on the ellipse with major and minor axes as 2 a and 2 b respectively, is equal to


68) A man running round a race course notes that the sum of the distance of two flag posts from him is always 10 m and the distance between the flag posts is 8 m. The area of the path he encloses ( in square meter ) is


69) If a bar of given length moves with its extremities on two fixed straight lines at right angles, then the locus of any point on bar marked on the bar describes a / an


70) The eccentricity of the ellipse is


71) Equation of auxiliary circle of ellipse is


72) Length of latusrectum of the ellipse is


73) In an ellipse, the distance between its foci is 6 and minor axis is 8. Then, its eccentricity is


74) The eccentricity of an ellipse, with centre at the origin, is 1 / 2. if one directrix as x = 4, the equation of the ellipse is


75) A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is 1/2, then length of semi-major axis is


76) The eccentricity of the hyperbola whose latusrectum is 8 and conjugate axis is equal to half of the distance between the foci is


77) The equation of the hyperbola whose foci are ( 6, 4 ) and ( - 4, 4 ) and eccentricity 2, is


78) The asymptotes of the hyperbola xy = hx + ky are


79) Eccentricity of hyperbola whose asymptotes are 3x - 4y = 7 and 4x + 3y = 8, is


80) A hyperbola has the asymptotes x + 2y = 3 and x - y = 0 and passes through ( 2, 1 ). Its centre is


81) The length of the transverse axis of the rectangular hyperbola xy = 18 is


82) The normal at P to a hyperbola of eccentricity e, intersects its transverse and conjugate axes at L and M respectively. If locus of the mid-point of L M is hyperbola, then eccentricity of the hyperbola is


83) If L is the chord of contact of the hyperbola H, then the equation of the corresponding pair of tangles is


84) If R is the point of intersection of the tangles of H at the extremities of the chord L, then equation of the chord of contact of R with respect to the parabola P is


85) STATEMENT I Asymptotes of hyperbola 3x + 4y = 2 and 4x - 3y = 5 are bisectors of transverse and conjugate axes of hyperbola.
STATEMENT II transverse and conjugate axes of hyperbola of the asymptotes.


86) If the focus of a parabola divides a focal chord of
the parabola in segments of length 3 and 2, the
length of the latus–rectum of the parabola is:


87) The locus of a point, which moves, such that the
sum of the squares of its distances from three
vertices of a triangle is constant is :


88) A circle of radius 5 units touches both the axes
and lies in the first quadrant. If the circle makes
one complete roll on x–axis along the positive
direction of x–axis, then its equation in new
position is :


89) A man running round a race course notes that the sum of the distances of two flag posts from him is always 10 metres and the distance between the flag posts is 8 metres. The area of the path he encloses in square metres is:


90) If the focal distance of an end of the minor axis of any ellipse (referred to its axes as the axis of x and y respectively) is k and the distance between the foci is 2h, then its equation is:


91) In a model, it is shown that an arc of a bridge is semi-elliptical having major axis horizontal. If the length of the base is 9 m and the highest point of the bridge is 3 m from the horizontal; the best approximation of the height of the arch, 2 m from the centre of the base is:


92) Chords of an ellipse are drawn through the positive end of the minor axis. Then their middle point lies on:


93) An ellipse has OB as semi–minor axis, F and F' its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is:


94) Locus of the centre of a circle which touches given circle externally is:


95) PM is perpendicular from a point on a rectangular hyperbola to its asymptotes, then the locus of the mid–point of PM is:


96) A circle touches the x–axis and also touches the circle with centre (0, 3) and radius 2. The locus of the centre of the circle is:


97) Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2 CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD
touching all the sides, then its radius is :


98) The circle passing through (1, –2) and touching
the axis of x at (3,0) also passesthrough the point:


99) The equation of the circle passing through the
points (1,0) and (0, 1) and having smallest radius
is :


100) Two vertices of an equilateral triangle are (-1,0) and (1,0) and its third vertex lies above the x-axis. The equation of its circumcircle, is


101) In an ellipse the distance between the foci is 8 and the distance between the directrices is 25. The length of major axis is


102) If the latusrectum of a hyperbola forms an equilateral triangle with the vertex at the center of the hyperbola, then the eccentricity of the hyperbola is


103) The locus of a point which moves such that the difference of its distances from two fixed points is always a constant, is


104) The normal at P to a hyperbola of eccentricity e, intersects its transverse and conjugate axes at L and M respectively. If locus of the mid point of LM is hyperbola, then eccentricity of the hyperbola is


105) An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms. are


106) ABCD is a square whose side is a. The equation of the circle circumscribing the square, taking AB and AD as axes of reference, is


107) Circles are drawn through the point (2,0) to cut intercepts of length 5 units on the x-axis. If their centres lie in the first quadrant, then their equation is


108) If the vertex and focus of a parabola are (3,3) and (-3,3) respectively, then its equation is


109) If the length of the major axis of an ellipse is three times the length of its minor axis, its eccentricity, is


110) A straight rod of length 9 units with its ends A,B always on x and y axes respectively. then, the locus of the centroid of ∆ OAB, is


111) S and T are the foci of an ellipse and B is end point of the minor axis . If STB is an equilateral triangle, the eccentricity of the ellipse is


112) The eccentricity of the hyperbola can never be equal to


113) The equation of the circle passing through (4, 5) and having the center (2, 2), is


114) The equation of the parabola whose vertex is at (2,-1) and focus at (2,-3) is


115) The line segment joining the points (4, 7) and (-2,-1) is a dismeter of a circle. If the circle intersects the x-axis at A and B, then AB is equal to


116) ABCDIs a square whose side isa. If AB and AD are axes of coordinates, the equation of the circle circumscribing the square will be


117) If (-4,3) and (8,3) are the vertices of an ellipse whose eccentricity is 5/6 then the equation of the ellipse is


118) If the coordinates of the center, a focus and adjacent vertex are (2,-3),(3,-3) and (4,-3) respectively, then the equation of the ellipse is


119) If the vertices of an ellipse are (-12,4) and (14,4) and eccentricity 12/13, then the equation of the ellipse is


120) The equation of the mirror that can reflect all incident rays from origin parallel to y-axis is


121) If the tangent and normal at any point P of a parabola meet the axes in Tand G respectively, then


122) Equation of the circle through the origin and making intercepts of 3 and 4 on the positive sides of the axes is


123) The parametric representation of a point of the ellipse whose foci are (3, 0) and (-1,0) and eccentricity 2/3 is


124) Which of the following equations gives circle?


125) If the vertex of a parabola is (0,2) and the extremities of latusrectum are (-6,4) and (6,4), then, its equation is


126) If the focus and vertex of a parabola are the points (0,2) and (0,4) respectively, then its equation is


127) In the standard form of an ellipse sum of the focal distances of a point is


128) The equation of the hyperbola whose foci are (6,5),(-4,5) and eccentricity 5/4, is


129) The difference of the focal distances of any point on the hyperbola is equal to its


130) Equation of a common tangent with positive slope to the circle as well as to the hyperbola is


131) All ellipse has its center at (1,-1) and semi-major axis = 8 and it passes through the point (1,3). The equation of the ellipse is


132) The area of the circle whose center is at (2, 3) and passing through (4, 6), is


133) If distance between directrices of a rectangular hyperbola is 10, then distance between its foci will be


134) The equation of the parabola with vertex (-1,1) and focus (2,1) is


135) Equation of the ellipse whose foci are (2,2) and (4,2) and the major axis is of length 10 is


136) The eccentricity of a rectangular hyperbola is


137) The locus of the poles of the focal chords of a parabola is …. of the parabola


138) Suppose a circle passes through (2, 2) and (9, 9) and touches the x-axis atP. If O is the origin, then OP is equal to


139) The two ends of latusrectum of a parabola are the points (3,6) and (-5,6). The focus is


140) A variable circle passes through the fixed point (2,0) and touches y-axis. Then, the locus of its center is


141) Equation of the parabola with its vertex at (1,1) and focus (3,1) is


142) A line is at a constant distance c from the origin and meets the coordinate axes in A and B. The locus of the centre of the circle passing through O,A,B is


143) For an ellipse with eccentricity 1/2 the centre is at the origin. If one directrix is x = 4, then the equation of the ellipse is


144) If (0,6)and (0,3) are respectively the vertex and focus of a parabola, then its equation is


145) A man running round a race course notes that the sum of the distances of two flag posts from him is always 10m and the distance between the flag posts is 8m. The area of the path he encloses (in square meter) is


146) The points (5,-7) lies outside the circle


147) If A,A' are the vertices, S,S' are the foci and Z,Z' are the feet of the directrices of an ellipse with centre C, then CS,CA,CZ are in


148) The length of the latusrectum of an ellipse is one third of its major axis. Its eccentricity would be


149) A circle touches y-axis at (0,2) and has an intercept of 4 units on the positive side of x-axis. The equation of the circle is


150) The equation of the hyperbola of given transverse axis 2a with its vertex mid-way between the centre and the corresponding focus is


151) The equation of the ellipse whose one focus is at (4,0) and whose eccentricity is 4/5 is


152) The radical axis of the coaxial system of circles with limiting points (1, 2) and (-2,1)is


153) The equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8, is


154) The equation of circle which touches the x-axis and y-axis at the points (1, 0) and (0, 1) respectively, is


155) Coordinates of foci of hyperbola are (-5,3) and (7, 3) and eccentricity is 3/2. Then ,length of its latusrectum is


156) The one which does not represent a hyperbola is


157) A circle passes through (0,0),(a,0) and (0,b) the coordinates of its centre are


158) If in a hyperbola, the distance between the foci is 10 and the transverse axis has length 8, then the length of its latusrectum is


159) Extremities of a diagonal of a rectangle are (0 ,0) and (4 ,3). The equations of the tangents to the circumcircle of the rectangle which are parallel to the diagonal are


160) If one end of the diameter is (1, 1) and the other end lies on the line x + y = 3, then locus of centre of circle is


161) The equation of the smallest circle passing through the points (2, 2) and (3, 3) is


162) The equation of the circle of radius 5 and touching the coordinate axes in third quadrant is


163) The center of the circle passing through (0 ,0) (a ,0) and (0 ,b) is


164) The equation of the parabola with its vertex at the origin, axis on the y-axis and passing through the point (6,-3) is


165) The sum of focal distance of any point on the ellipse with major and minor axes as 2a and 2b respectively, is equal to


166) The length of major and minor axis of an ellipse are 10 and 8 respectively and its major axis along y-axis the equation of the ellipse referred to its center as origin is


167) The equation of two circles which touch the y-axis at (0, 3) and make an intercept of 8 unit on x-axis, are


168) If the points (2, 0), (0, 1), (4, 5) and (0,c) are concyclic, then the value of c is


169) The equation of the circle passing through (0,0) and belonging to the system of circles of which (3, 1) and (-1, 5) are limiting points, is


170) The equations of the circle which pass through the origin and makes intercepts of lengths 4 and 8 on the x and y-axes respectively are


171) The parametric representation of a point on the ellipse whose foci are (-1,0) and (7,0) and eccentricity 1/2 is


172) Four distinct points (2k, 3k),(1 ,0),(0 ,1) and (0,0) lie on a circle for


173) The equation of family of circles with center at (h,k) touching the x-axis is given by


174) The circle on focal radii of a parabola as diameter touches the


175) A set of points is such that each point is three times as far away from the y-axis as it is from the point (4,0). Then, the locus of the points is


176) If transverse and conjugate axes of hyperbola are equal then it’s eccentricity is


177) Distance between foci is 8 and distance between directrices is 6 of hyperbola, then length of latus rectum is


178) The number of maximum normals which can be drawn from a point to ellipse is


179) he product of perpendiculars drawn from any point of a hyperbola to its asymptotes is


180) The equation of the ellipse whose distance between foci is equal to 8 and distance between the directrix is 18, is


181) AB is a diameter of a circle and C is any point on the circumference of the circle. Then,


182) The area of the circle centered at (1, 2) and passing through (4, 6), is