JEE Mathematics Ellipse MCQs Set 03

Question. If distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is
(a) \( \frac{1}{2} \)
(b) \( \frac{2}{3} \)
(c) \( \frac{1}{\sqrt{3}} \)
(d) \( \frac{4}{5} \)
Answer: (c) \( \frac{1}{\sqrt{3}} \)

Question. If the eccentricity of an ellipse be \( 5/8 \) and the distance between its foci be 10, then its latus rectum is
(a) \( \frac{39}{4} \)
(b) 12
(c) 15
(d) \( \frac{37}{2} \)
Answer: (a) \( \frac{39}{4} \)

Question. The curve represented by \( x = 3(\cos t + \sin t), y = 4(\cos t - \sin t) \), is
(a) ellipse
(b) parabola
(c) hyperbola
(d) circle
Answer: (a) ellipse

Question. If the distance of a point on the ellipse \( \frac{x^2}{6} + \frac{y^2}{2} = 1 \) from the centre is 2, then the eccentric angle is
(a) \( \pi/3 \)
(b) \( \pi/4 \)
(c) \( \pi/6 \)
(d) \( \pi/2 \)
Answer: (b) \( \pi/4 \)

Question. An ellipse having foci at \( (3, 3) \) and \( (-4, 4) \) and passing through the origin has eccentricity equal to
(a) \( \frac{3}{7} \)
(b) \( \frac{2}{7} \)
(c) \( \frac{5}{7} \)
(d) \( \frac{3}{5} \)
Answer: (c) \( \frac{5}{7} \)

Question. A tangent having slope of \( -\frac{4}{3} \) to the ellipse \( \frac{x^2}{18} + \frac{y^2}{32} = 1 \) intersects the major & minor axes in points A & B respectively. If C is the centre of the ellipse then the area of the triangle ABC is
(a) 12 sq. units
(b) 24 sq. units
(c) 36 sq. units
(d) 48 sq. units
Answer: (b) 24 sq. units

Question. The equation to the locus of the middle point of the portion of the tangent to the ellipse \( \frac{x^2}{16} + \frac{y^2}{9} = 1 \) included between the co-ordinate axes is the curve
(a) \( 9x^2 + 16y^2 = 4x^2y^2 \)
(b) \( 16x^2 + 9y^2 = 4x^2y^2 \)
(c) \( 3x^2 + 4y^2 = 4x^2y^2 \)
(d) \( 9x^2 + 16y^2 = x^2y^2 \)
Answer: (a) \( 9x^2 + 16y^2 = 4x^2y^2 \)

Question. An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is
(a) \( \sqrt{3} \)
(b) 2
(c) \( 2\sqrt{2} \)
(d) \( \sqrt{5} \)
Answer: (b) 2

Question. Which of the following is the common tangent to the ellipses \( \frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1 \) & \( \frac{x^2}{a^2} + \frac{y^2}{a^2 + b^2} = 1 \)?
(a) \( ay = bx + \sqrt{a^4 - a^2b^2 + b^4} \)
(b) \( by = ax - \sqrt{a^4 + a^2b^2 + b^4} \)
(c) \( ay = bx - \sqrt{a^4 + a^2b^2 + b^4} \)
(d) \( by = ax - \sqrt{a^4 - a^2b^2 + b^4} \)
Answer: (b) \( by = ax - \sqrt{a^4 + a^2b^2 + b^4} \)

Question. Angle between the tangents drawn from point \( (4, 5) \) to the ellipse \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \) is
(a) \( \pi/3 \)
(b) \( 5\pi/6 \)
(c) \( \pi/4 \)
(d) \( \pi/2 \)
Answer: (d) \( \pi/2 \)

Question. The eccentricity of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \) is decreasing at the rate of 0.1/second due to change in semi minor axis only. The time at which ellipse become auxiliary circle is
(a) 2 seconds
(b) 3 seconds
(c) 4 seconds
(d) 5 seconds
Answer: (d) 5 seconds

Question. The point of intersection of the tangents at the point P on the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), and its corresponding point Q on the auxiliary circle meet on the line
(a) \( x = a/e \)
(b) \( x = 0 \)
(c) \( y = 0 \)
(d) None of the options
Answer: (c) \( y = 0 \)

Question. Q is a point on the auxiliary circle of an ellipse. P is the corresponding point on ellipse. N is the foot of perpendicular from focus S, to the tangent of auxiliary circle at Q. Then
(a) \( SP = SN \)
(b) \( SP = PQ \)
(c) \( PN = SP \)
(d) \( NQ = SP \)
Answer: (a) \( SP = SN \)

Question. Q is a point on the auxiliary circle corresponding to the point P of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). If T is the foot of the perpendicular dropped from the focus S onto the tangent to the auxiliary circle at Q then the \( \Delta SPT \) is
(a) isosceles
(b) equilateral
(c) right angled
(d) right isosceles
Answer: (a) isosceles

Question. The equation of the normal to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at the positive end of latus rectum is
(a) \( x + ey + e^2a = 0 \)
(b) \( x - ey - e^3a = 0 \)
(c) \( x - ey - e^2a = 0 \)
(d) None of the options
Answer: (b) \( x - ey - e^3a = 0 \)

Question. The eccentric angle of the point where the line, \( 5x - 3y = 8\sqrt{2} \) is a normal to the ellipse \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \) is
(a) \( 3\pi/4 \)
(b) \( \pi/4 \)
(c) \( \pi/6 \)
(d) \( \tan^{-1} 2 \)
Answer: (b) \( \pi/4 \)

Question. PQ is a double ordinate of the ellipse \( x^2 + 9y^2 = 9 \), the normal at P meets the diameter through Q at R, then the locus of the mid point of PR is
(a) a circle
(b) a parabola
(c) an ellipse
(d) a hyperbola
Answer: (c) an ellipse

Question. The equation of the chord of the ellipse \( 2x^2 + 5y^2 = 20 \) which is bisected at the point \( (2, 1) \) is
(a) \( 4x + 5y + 13 = 0 \)
(b) \( 4x + 5y = 13 \)
(c) \( 5x + 4y + 13 = 0 \)
(d) \( 4x + 5y = 13 \)
Answer: (b) \( 4x + 5y = 13 \)

Question. If \( F_1 \) & \( F_2 \) are the feet of the perpendiculars from the foci \( S_1 \) & \( S_2 \) of an ellipse \( \frac{x^2}{5} + \frac{y^2}{3} = 1 \) on the tangent at any point P on the ellipse, then \( (S_1F_1) \cdot (S_2F_2) \) is equal to
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (b) 3

Question. If \( \tan \theta_1 \cdot \tan \theta_2 = - \frac{a^2}{b^2} \) then the chord joining two points \( \theta_1 \) & \( \theta_2 \) on the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) will subtend a right angle at
(a) focus
(b) centre
(c) end of the major axis
(d) end of the minor axis
Answer: (b) centre

Question. \( x - 2y + 4 = 0 \) is a common tangent to \( y^2 = 4x \) & \( \frac{x^2}{4} + \frac{y^2}{b^2} = 1 \). Then the value of \( b \) and the other common tangent are given by
(a) \( b = \sqrt{3} ; x + 2y + 4 = 0 \)
(b) \( b = 3; x + 2y + 4 = 0 \)
(c) \( b = \sqrt{3} ; x + 2y - 4 = 0 \)
(d) \( b = \sqrt{3} ; x - 2y - 4 = 0 \)
Answer: (a) \( b = \sqrt{3} ; x + 2y + 4 = 0 \)

Question. The tangent at any point \( P \) on a standard ellipse with foci as \( S \) & \( S' \) meets the tangents at the vertices \( A \) & \( A' \) in the points \( V \) & \( V' \), then
(a) \( \ell(AV) \cdot \ell(A'V') = b^2 \)
(b) \( \ell(AV) \cdot \ell(A'V') = a^2 \)
(c) \( \angle V'SV = 90^\circ \)
(d) \( VS'V'S \) is a cyclic quadrilateral
Answer: (a) \( \ell(AV) \cdot \ell(A'V') = b^2 \), (c) \( \angle V'SV = 90^\circ \), (d) \( VS'V'S \) is a cyclic quadrilateral

Question. The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is \( \pi/4 \) is
(a) \( \frac{(a^2 - b^2)ab}{a^2 + b^2} \)
(b) \( \frac{(a^2 + b^2)ab}{a^2 - b^2} \)
(c) \( \frac{ab(a^2 + b^2)}{(a^2 - b^2)} \)
(d) \( \frac{(a^2 - b^2)ab}{(a^2 - b^2)ab} \)
Answer: (a) \( \frac{(a^2 - b^2)ab}{a^2 + b^2} \)

Question. An ellipse is such that the length of the latus rectum is equal to the sum of the lengths of its semi principal axes. Then
(a) Ellipse becomes a circle
(b) Ellipse becomes a line segment between the two foci
(c) Ellipse becomes a parabola
(d) None of the options
Answer: (a) Ellipse becomes a circle

Question. The line, \( lx + my + n = 0 \) will cut the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) in points whose eccentric angles differ by \( \pi/2 \) if
(a) \( a^2l^2 + b^2n^2 = 2m^2 \)
(b) \( a^2m^2 + b^2l^2 = 2n^2 \)
(c) \( a^2l^2 + b^2m^2 = 2n^2 \)
(d) \( a^2n^2 + b^2m^2 = 2l \)
Answer: (c) \( a^2l^2 + b^2m^2 = 2n^2 \)

Question. A circle has the same centre as an ellipse & passes through the foci \( F_1 \) & \( F_2 \) of the ellipse, such that the two curves intersect in 4 points. Let 'P' be any one of their point of intersection. If the major axis of the ellipse is 17 & the area of the triangle \( PF_1F_2 \) is 30, then the distance between the foci is
(a) 11
(b) 12
(c) 13
(d) None of the options
Answer: (c) 13

Question. The normal at a variable point \( P \) on an ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) of eccentricity \( e \) meets the axes of the ellipse in \( Q \) and \( R \) then the locus of the mid-point of \( QR \) is a conic with an eccentricity \( e' \) such that
(a) \( e' \) is independent of \( e \)
(b) \( e' = 1 \)
(c) \( e' = e \)
(d) \( e' = 1/e \)
Answer: (c) \( e' = e \)

Question. The length of the normal (terminated by the major axis) at a point of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is
(a) \( \frac{b}{a}(r + r_1) \)
(b) \( \frac{b}{a}|r - r_1| \)
(c) \( \frac{b}{a}\sqrt{rr_1} \)
(d) independent of \( r, r_1 \)
where \( r \) and \( r_1 \) are the focal distance of the point.
Answer: (c) \( \frac{b}{a}\sqrt{rr_1} \)

Question. Point 'O' is the centre of the ellipse with major axis \( AB \) and minor axis \( CD \). Point \( F \) is one focus of the ellipse. If \( OF = 6 \) and the diameter of the inscribed circle of triangle \( OCF \) is 2, then the product \( (AB)(CD) \) is equal to
(a) 65
(b) 52
(c) 78
(d) None of the options
Answer: (a) 65

Question. If \( P \) is a point of the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), whose focii are \( S \) and \( S' \). Let \( \angle PSS' = \alpha \) and \( \angle PS'S = \beta \), then
(a) \( SP + PS' = 2a \), if \( a > b \)
(b) \( PS + PS' = 2b \), if \( a < b \)
(c) \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2} = \frac{1-e}{1+e} \)
(d) \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2} = \frac{\sqrt{a^2 - b^2}}{b^2} [a - \sqrt{a^2 - b^2}] \) when \( a > b \)
Answer: (a) \( SP + PS' = 2a \), if \( a > b \), (b) \( PS + PS' = 2b \), if \( a < b \), (c) \( \tan \frac{\alpha}{2} \tan \frac{\beta}{2} = \frac{1-e}{1+e} \)

Question. If the chord through the points whose eccentric angles are \( \theta \) & \( \phi \) on the ellipse, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) passes through the focus, then the value of \( \tan(\theta/2) \tan(\phi/2) \) is
(a) \( \frac{e + 1}{e - 1} \)
(b) \( \frac{e - 1}{e + 1} \)
(c) \( \frac{1 - e}{1 + e} \)
(d) \( \frac{1 + e}{1 - e} \)
Answer: (a) \( \frac{e + 1}{e - 1} \), (b) \( \frac{e - 1}{e + 1} \)

Question. If \(x_1\), \(x_2\), \(x_3\) as well as \(y_1\), \(y_2\), \(y_3\) are in G.P. with the same common ratio, then the points \((x_1, y_1), (x_2, y_2) \text{ & } (x_3, y_3)\)
(a) lie on a straight line
(b) lie on an ellipse
(c) lie on a circle
(d) are vertices of a triangle.
Answer: (a) lie on a straight line

Question. On the ellipse, \(4x^2 + 9y^2 = 1\), the points at which the tangents are parallel to the line \(8x = 9y\) are
(a) \( \left( \frac{2}{5}, \frac{1}{5} \right) \)
(b) \( \left( \frac{2}{5}, -\frac{1}{5} \right) \)
(c) \( \left( -\frac{2}{5}, -\frac{1}{5} \right) \)
(d) \( \left( \frac{2}{5}, -\frac{1}{5} \right) \)
Answer: (b) \( \left( \frac{2}{5}, -\frac{1}{5} \right)

Question. The area of the quadrilateral formed by the tangents at the ends of the latus rectum of the ellipse \( \frac{x^2}{9} + \frac{y^2}{5} = 1 \) is
(a) \( 9\sqrt{3} \) sq. units
(b) \( 27\sqrt{3} \) sq. units
(c) 27 sq. units
(d) None of the options
Answer: (c) 27 sq. units

Question. The value of \(\theta\) for which the sum of intercept on the axis by the tangent at the point \((3\sqrt{3} \cos \theta, \sin \theta), 0 < \theta < \pi/2\) on the ellipse \( \frac{x^2}{27} + y^2 = 1 \) is least, is
(a) \( \pi/6 \)
(b) \( \pi/4 \)
(c) \( \pi/3 \)
(d) \( \pi/8 \)
Answer: (a) \( \pi/6 \)

Question. The locus of the middle point of the intercept of the tangents drawn from an external point to the ellipse \(x^2 + 2y^2 = 2\), between the coordinates axes, is
(a) \( \frac{1}{x^2} + \frac{1}{2y^2} = 1 \)
(b) \( \frac{1}{4x^2} + \frac{1}{2y^2} = 1 \)
(c) \( \frac{1}{2x^2} + \frac{1}{4y^2} = 1 \)
(d) \( \frac{1}{2x^2} + \frac{1}{y^2} = 1 \)
Answer: (c) \( \frac{1}{2x^2} + \frac{1}{4y^2} = 1 \)

Question. The minimum area of triangle formed by the tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) and coordinate axes is
(a) ab sq. units
(b) \( \frac{a^2 + b^2}{2} \) sq. units
(c) \( \frac{(a + b)^2}{2} \) sq. units
(d) \( \frac{a^2 + ab + b^2}{3} \) sq. units
Answer: (a) ab sq. units

Question. Let \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), \(y_1 < 0, y_2 < 0\), be the end points of the latus rectum of the ellipse \(x^2 + 4y^2 = 4\). The equations of parabolas with latus rectum PQ are
(a) \( x^2 + 2\sqrt{3}y = 3 + \sqrt{3} \)
(b) \( x^2 - 2\sqrt{3}y = 3 + \sqrt{3} \)
(c) \( x^2 + 2\sqrt{3}y = 3 - \sqrt{3} \)
(d) \( x^2 - 2\sqrt{3}y = 3 - \sqrt{3} \)
Answer: (b) \( x^2 - 2\sqrt{3}y = 3 + \sqrt{3} \)

Question. The line passing through the extremity A of the major axis of extremity B of the minor axis of the ellipse \(x^2 + 9y^2 = 9\) meets the auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is
(a) 31/10
(b) 29/10
(c) 21/10
(d) 27/10
Answer: (d) 27/10

Question. The normal at a point P on the ellipse \(x^2 + 4y^2 = 16\) meets the x-axis at Q. If M is the midpoint of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points
(a) \( \left( \pm \frac{3\sqrt{5}}{2}, \pm \frac{2}{7} \right) \)
(b) \( \left( \pm \frac{3\sqrt{5}}{2}, \pm \frac{\sqrt{19}}{4} \right) \)
(c) \( \left( \pm 2\sqrt{3}, \pm \frac{1}{7} \right) \)
(d) \( \left( \pm 2\sqrt{3}, \pm \frac{4\sqrt{3}}{7} \right) \)
Answer: (c) \( \left( \pm 2\sqrt{3}, \pm \frac{1}{7} \right) \)

Question. The coordinates of A and B are
(a) (3, 0) and (0, 2)
(b) \( \left( \frac{8}{5}, \frac{2\sqrt{161}}{15} \right) \) and \( \left( -\frac{9}{5}, \frac{8}{5} \right) \)
(c) \( \left( -\frac{8}{5}, \frac{2\sqrt{161}}{15} \right) \) and (0, 2)
(d) (3, 0) and \( \left( -\frac{9}{5}, \frac{8}{5} \right) \)
Answer: (d) (3, 0) and \( \left( -\frac{9}{5}, \frac{8}{5} \right) \)

Question. The orthocenter of the triangle PAB is
(a) \( \left( 5, \frac{8}{7} \right) \)
(b) \( \left( \frac{7}{5}, \frac{25}{8} \right) \)
(c) \( \left( \frac{11}{5}, \frac{8}{5} \right) \)
(d) \( \left( \frac{8}{25}, \frac{7}{5} \right) \)
Answer: (c) \( \left( \frac{11}{5}, \frac{8}{5} \right) \)

Question. The equation of the locus of the point whose distances from the point P and the line AB are equal is
(a) \( 9x^2 + y^2 - 6xy - 54x - 62y + 241 = 0 \)
(b) \( x^2 + 9y^2 + 6xy - 54x + 62y - 241 = 0 \)
(c) \( 9x^2 + 9y^2 - 6xy - 54x - 62y - 241 = 0 \)
(d) \( x^2 + y^2 - 2xy + 27x + 31y - 120 = 0 \)
Answer: (a) \( 9x^2 + y^2 - 6xy - 54x - 62y + 241 = 0 \)

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