Practice Questions

Q69.If the straight line 2x −3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β) , then β equals : (1) −5 (2) 353 (3) 5 (4) −353

Q69.The tangent to the parabola 𝑦2 = 4𝑥 at the point where it intersects the circle 𝑥2 + 𝑦2 = 5 in the first quadrant, passes through the point: (1) 1 3 (2) -1 4 4, 4 3, 3 1 1 3 7 (3) - 4, 2 (4) 4, 4

Q69.Let S be the set of all triangles in the xy -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is: (1) 36 (2) 32 (3) 9 (4) 18

Q69.If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is: (1) 120 (2) 60 13 13 13 13 (3) (4) 5 2

Q69.Three circles of radii 𝑎, 𝑏, 𝑐, 𝑎< 𝑏< 𝑐 touch each other externally. If they have 𝑥- axis as a common tangent, then: (1) 1 1 1 (2) 𝑎, 𝑏, 𝑐 are in A.P. √𝑎= √𝑏+ √𝑐 √𝑎, √𝑏, √𝑐 are in A.P. (3) √𝑏=1 √𝑎+1 √𝑐1 (4)

Q69.A point P moves on the line 2x −3y + 4 = 0. If Q(1, 4) and R(3, −2) are fixed points, then the locus of the centroid of ΔPQR is a line: (1) with slope 2 (2) with slope 3 3 2 (3) parallel to y-axis (4) parallel to x-axis

Q69.If the circles x2 + y2 + 5Kx + 2y + K = 0 and 2(x2 + y2) + 2Kx + 3y −1 = 0, (K ∈R), intersect at the points P and Q, then the line 4x + 5y −K = 0 , passes through P and Q, for: (1) exactly two values of K (2) no value of K (3) exactly one value of K (4) infinitely many values of K y2

Q69.A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (−8, 5) and (6, 5), then the area of the rectangle (in sq. units ) is: (1) 72 (2) 98 (3) 56 (4) 84

Q69.The length of the chord of the parabola x2 = 4y having equation x −√2y + 4√2 = 0 is JEE Main 2019 (10 Jan Shift 2) JEE Main Previous Year Paper (1) 6√3 units (2) 8√2 units (3) 2√11 units (4) 3√2 units y2 x2 = r ≠±1. Then S represents: y) ∈R2 : 1+r − 1−r

Q69.The sum of the squares of the lengths of the chords intercepted on the circle, 𝑥2 + 𝑦2 = 16, by the lines, 𝑥+ 𝑦= 𝑛, 𝑛∈𝑁, where 𝑁 is the set of all natural numbers is: (1) 210 (2) 105 (3) 320 (4) 160

Q70.If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13 , then the eccentricity of the hyperbola is: (1) 13 (2) 2 12 (3) 13 (4) 13 6 8

Q70.If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is (1) 24 (2) 25 (3) 22 (4) 20 , then a value of m is:

Q70.A circle touching the x− axis at (3, 0) and making an intercept of length 8 on the y− axis passes through the point: (1) (3, 10) (2) (2, 3) (3) (3, 5) (4) (1, 5)

Q70.Two circles with equal radii are intersecting at the points (0,1) and (0,-1) . The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is: (1) 1 (2) 2 (3) 2√2 (4) √2

Q70.Let C1 and C2 be the centres of the circles x2 + y2 −2x −2y −2 = 0 and x2 + y2 −6x −6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral PC1 QC2 is : JEE Main 2019 (12 Jan Shift 1) JEE Main Previous Year Paper (1) 6 (2) 4 (3) 8 (4) 9

Q70.Let 𝑂0,0 and 𝐴0,1 be two fixed points. Then, the locus of a point 𝑃 such that the perimeter of 𝛥𝐴𝑂𝑃 is 4 is (1) 8𝑥2 + 9𝑦2 - 9𝑦= 18 (2) 9𝑥2 - 8𝑦2 + 8𝑦= 16 (3) 8𝑥2 - 9𝑦2 + 9𝑦= 18 (4) 9𝑥2 + 8𝑦2 - 8𝑦= 16

Q70.Let the equations of two sides of a triangle be 3x −2y + 6 = 0 and 4x + 5y −20 = 0. If the orthocenter of this triangle is at (1, 1) then the equation of it's third side is: (1) 122y + 26x + 1675 = 0 (2) 26x −122y −1675 = 0 (3) 26x + 61y + 1675 = 0 (4) 122y −26x −1675 = 0

Q70.If the normal to the ellipse 3𝑥2 + 4𝑦2 = 12 at a point 𝑃 on it is parallel to the line, 2𝑥+ 𝑦= 4 and the tangent to the ellipse at 𝑃 passes through 𝑄( 4,4 ) then 𝑃𝑄 is equal to: (1) √61 (2) 5√5 2 2 (3) √157 (4) √221 2 2

Q70.The equation of a tangent to the parabola, x2 = 8y, which makes an angle θ with the positive direction of x− axis, is (1) y = xtanθ + 2cotθ (2) y = xtanθ −2cotθ (3) x = ycotθ + 2tanθ (4) x = ycotθ −2tanθ

Q70.Axis of a parabola lies along 𝑥-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive 𝑥-axis then which of following points does not lie on it? (1) 6, 4√2 (2) 5, 2√6 (3) 8, 6 (4) 4, - 4

Q70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1

Q70.The common tangent to the circles x2 + y2 = 4 and x2 + y2 + 6x + 8y −24 = 0 also passes through the point: (1) (4, −2) (2) (−4, 6) (3) (6, −2) (4) (−6, 4) JEE Main 2019 (09 Apr Shift 2) JEE Main Previous Year Paper

Q70.In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at 0,5√3, then the length of its latus rectum is: (1) 6 (2) 10 (3) 8 (4) 5

Q70.If a circle C passing through the point (4, 0) touches the circle x2 + y2 + 4x −6y = 12 externally at the point (1, −1), then the radius of C is: (1) 4 units (2) 5 units (3) 2√5 units (4) √57 units

Q70.If the line x −2y = 12 is a tangent to the ellipse x2 + = 1 at the point (3, −92 ), then the length of the a2 b2 latus rectum of the ellipse is (1) 5 units (2) 12√2 units (3) 9 units (4) 8√3 units 5x = 4√5