Download PTU B.Sc-Hons 2020 March 1st Sem 77313 Co Ordinate Geometry Question Paper

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Roll No. EEEEEEEEEEE Total No. of Pages : 02

Total No. of Questions : 09

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Bachelor of Science - Honours (Mathematics) (Sem.-1)

CO-ORDINATE GEOMETRY

Subject Code : UC-BSHM-102-19

M.Code : 77313

Time : 3 Hrs. Max. Marks : 60

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INSTRUCTIONS TO CANDIDATES :

1. SECTION-A is COMPULSORY consisting of TEN questions carrying TWO marks each.
2. SECTION -B & C have FOUR questions each.
3. Attempt any FIVE questions from SECTION B & C carrying EIGHT marks each.
4. Select atleast TWO questions from SECTION - B & C.

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SECTION-A

1. Solve the following :
   1. For what values of k does the equation 12x² — 10xy + 2y² + 11x — 5y + k = 0 represent two straight lines?
   2. Find the equations of the tangent and normal at the point of the parabola y² = 8x whose ordinate is 4.
   3. Find the pole of the line x — 2y + 3 = 0 w.r.t. the ellipse 3x² + 4y² = 12.
   4. If e₁ and e₂ be the eccentricities of a hyperbola and of the conjugate hyperbola, then show that 1/e₁² + 1/e₂² = 1.

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   5. Show that the equation r² — 2br cos (? — a) = c describes a circle with centre (b, a) if b²+c>0.
   6. Show that if ax² + 2hxy + by² = 1 and a'x² + 2h'xy + b'y² = 1 represent the same conic and the axes are rectangular, then (a — b)² +4h² = (a' — b')² + 4h'².
   7. Find the equation to a circle, the axis of coordinates being two straight lines through its centre at right angles.
   8. Find the equations of the tangents to the circle x² + y² — 6x + 4y = 12 which are parallel to the line 4x + 3y +5=0.
   9. Prove that the circles x² + y² — 2ax + c = 0 and x² + y² + 2by — c = 0 intersect orthogonally.

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   10. If pairs of straight lines x² — 2pxy — y² = 0 and x² — 2qxy — y² = 0 be such that each pair bisects the angle between the other pair, prove that pq =—1.

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SECTION-B

1. Find the angle between the straight lines given by the equation ax² + 2hxy + by² = 0. Also find the condition of perpendicularity.

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2. Find the equations of the lines joining the origin to the points of intersection of the line 2x — 3y + 4 = 0 with the curve x² + 4xy + 2y² + 12x + 4y = 0, and show that they are at right angles.
3. 1. Find the pole of the straight line 9x + y — 28 = 0 with respect to the circle 2x²+2y²—3x+5y—7=0.
   2. Find the lengths of the tangents drawn to the circle 3x² + 3y² — 7x — 6y = 12 from the point (6, -7).
4. Find the locus of a point P which is such that its polar with respect to one circle touches a second circle.

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SECTION-C

1. 1. Prove that the locus of the poles of the chords which are normal to the parabola y² = 4ax is the curve y² (x +2a) + 4a³ = 0.
   2. Prove that the chord of a parabola which subtends a right angle at the vertex meets its axis in a fixed point.
2. 1. If the normal at the end of a latus rectum of an ellipse passes through one extremity of the minor axis, show that the eccentricity of the curve is given by the equation e⁴+e²-1=0.

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   2. The chord of contact of tangents from a point P to the hyperbola x²/a² - y²/b² =1 subtends a right angle at the centre. Prove that locus of P is the ellipse x²/a⁴ + y²/b⁴ = 1/a² + 1/b².
3. 1. Through what angle should the axes be rotated so that the mixed term may disappear from the equation 17x² — 16xy + 17y² — 225 =0?
   2. On shifting the origin to the point (1, —1), the axis remaining parallel to the original axis, the equation of a curve becomes 4x² + y² + 3x — 4y + 2 = 0. Find its original equation.

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4. 1. Identify the curve represented by the equation 3x² + 2xy + 3y² + 18x + 22y + 50 = 0. Reduce it to standard form by suitable transformation of axes.
   2. Find the equation of the chord of contact of tangents drawn from the point (r₁, ?₁) to the conic l/r = 1+ecos?.

NOTE : Disclosure of Identity by writing Mobile No. or Making of passing request on any page of Answer Sheet will lead to UMC against the Student.

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