Parabola practice · Mixed difficulty

Option B ($xy=a$) is correct because the condition for tangency to the parabola $y^2=4ax$ gives the relation $\alpha \beta = a$, leading to the locus $xy=a$.

Option A (8) is correct because it is the length of the focal chord calculated using the given angle and parabola equation.

Option A (A parabola) is correct because the equation represents the set of points equidistant from a point (2,3) and a line 3x + 4y - 2 = 0, which is the definition of a parabola.

The correct answer is C ((3,4),(-13,4)) because these are the coordinates of the endpoints of the latus rectum after transforming the given equation into standard parabolic form.

Option B ($x-y-1=0$) is correct because it represents the line joining the foci of the given parabolas, calculated by finding the coordinates of the foci and then the equation of the line.

Option A (3) is correct because the value of 'a' in the directrix equation is 3, derived from the given parabola equation.

Option B ($9 x^2=12 y+8$) is correct because it represents the locus of point Q derived from the given conditions of the parabola and ratio division.

Option A (Vertex is $(2 a / 3,0)$) is correct because the locus of the centroid is a parabola with vertex at $(2a/3, 0)$.

Option D ($y^2-4 x+2=0$) is correct because it represents the locus of the midpoint of PQ, derived by substituting the midpoint coordinates into the equation of the parabola.

Option C ($a^2 \geq 8 b^2$) is correct because the condition for the tangents to the parabola $y^2=4ax$ to be normals to the parabola $x^2=4by$ leads to this inequality based on the discriminant of a quadratic equation.

Option A ($(1 / 4,1 / 2)$) is correct because the derivative of the curve $y^2 = x$ gives the slope of the tangent, which is equal to $\tan(45^{\circ}) = 1$.

Option C ($a \cos \alpha+p=0$) is correct because it represents the condition for the line $x \cos \alpha+y \sin \alpha=p$ to touch the parabola $y^2=4 a(x+a)$.

Option C ($\left( {2, \pm 4} \right)$) is correct because it's the point on the parabola with the given focal distance, calculated using the definition of focal distance.

Option A ($3x+4y+1=0$) is correct because it represents the directrix derived from the given parabola equation by comparing it to the standard form.

Option A ($y^2=12 x$) is correct because it satisfies the definition of a parabola with focus (3,0) and directrix x+3=0.

Option A (2) is correct because finding the locus of the midpoint of the intercepts of a tangent to the parabola gives the correct value of lambda.

Option C (Zero) is correct because the point (1, -2) lies inside the parabola, and no real tangents can be drawn from an interior point.

Option A (3) is correct because solving for the slopes of the tangents and then applying the tangent formula yields tan α = 3.

Option C (y=(x-1)) is correct because using the tangent formula and the given condition $Q_1 + Q_2 = 45^{\circ}$, we can derive the locus of point P as y = x-1.

Option C (${(y - 1)^2} = 8(x - 1)$) is correct because it represents a parabola with the given vertex and focus, opening to the right.

Option C ($\left( {1,0} \right)$) is correct because the vertex of a parabola is the midpoint between the focus and the directrix.

The correct answer is B because the locus of the midpoints of chords with slope m for the parabola $x^2 = 4py$ is a horizontal line at $y = \frac{{x^2}}{{4p}}$.

Option C ((0,0)) is correct because the radical center of circles drawn on focal chords as diameters is always at the origin.

Option C (${{\left( {y - 1} \right)}^2} = 8\left( {x - 1} \right)$) is correct because it represents the equation of a parabola with the given vertex and focus.

Option A ($x + 4 = 0$) is correct because it is found by determining the value of 'a' from the focal chord and then using the formula for the directrix of the parabola.

Option C ($\frac{{\sqrt{3}}}{{2}}$) is correct because it is the value obtained after calculating the angle of intersection between the two parabolas.

Option A (2) is correct. The length of the latus rectum is twice the perpendicular distance from the focus to the directrix.

Option B ($\frac{{125}}{{16}}$) is correct because it is the square of the distance between the point P on the parabola and the point Q where the normal at P intersects the parabola again, calculated using the parabola's equation and normal properties.

Option D (any $a>0$) is correct because the condition for collinearity of A, B, and S holds true regardless of the value of 'a'.

Option A (${p^2} + pq + 2 = 0$) is correct because it represents the relationship between the parameters p and q when the normal at one point on the parabola intersects the curve again at another point.