Mathematics : Conic Section

Let P be the mid-point of a chord joining the vertex of the parabola y² = 8x to another point on it. Then, the locus of P is

• y² = 2x

• y² = 4x

• $\frac{{\mathrm{x}}^2}{4} + {\mathrm{y}}^2 = 1$

• ${\mathrm{x}}^2 +\hspace{0.17em}\frac{{\mathrm{y}}^2}{4}$ = 1

The line x = 2y intersects the ellipse $\frac{{\mathrm{x}}^2}{4} + {\mathrm{y}}^2 = 1$ at the points P and Q. The equation of the circle with PQ as diameter is

• ${\mathrm{x}}^2 + {\mathrm{y}}^2 = \frac{1}{2}$

• x² + y² = 1

• x² + y² = 2

• x² + y² = $\frac{5}{2}$

The eccentric angle in the first quadrant of a point on the ellipse $\frac{{\mathrm{x}}^2}{10} + \frac{{\mathrm{y}}^2}{8} = 1$  at a distance units from the centre of the ellipse is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The transverse axis of a hyperbola is along the x - axis and its length is 2a. The vertex of the hyperbola bisects the line segment joining the centre and the focus. The equation of the hyperbola is

• 6x² - y² = 3a²

• x² - 3y² = 3a²

• x² - 6y² = 3a²

• 3x² - y² = 3a²

A point moves in such a way that the difference of its distance from two points (8, 0) and (- 8, 0) always remains 4. Then, the locus of the point is

• a circle

• a parabola

• an ellipse

• a hyperbola

Let C₁ and C₂ denote the centres of the circles x² + y² = 4 and (x - 2)² + y² = 1 respectively and let P and Q be their points of intersection. Then, the areas of $∆$C₂PQ and CPQ are in the ratio

• 3 : 1

• 5 : 1

• 7 : 1

• 9 : 1

The incentre of an equilateral triangle is (1, 1) and the equation of one side is 3x + 4y + 3 = 0. Then, the equation of the circumcircle of the triangle is

• x² + y² - 2x - 2y - 2 = 0

• x² + y² - 2x - 2y - 14 = 0

• x² + y² - 2x - 2y + 2 = 0

• x² + y² - 2x - 2y + 14 = 0

The eccentricity of the hyperbola 4x² - 9y² = 36 is

• $\frac{\sqrt{11}}{3}$

• $\frac{\sqrt{15}}{3}$

• $\frac{\sqrt{13}}{3}$

• $\frac{{\displaystyle \sqrt{14}}}{{\displaystyle 3}}$

The length of the latus rectum of the ellipse 16x² + 25y² = 400 is

• 5/16 unit

• 32/5 unit

• 16/5 unit

• 5/32 unit

The vertex of the parabola y² + 6x - 2y + 13 = 0 is

• (1, - 1)

• (- 2, 1)

• $\left(\frac{3}{2}, 1\right)$

• $\left(- \frac{7}{2}, 1\right)$