Parabola practice · Medium

Option B (y = 1) is correct because by eliminating the parameter t, we get the parabola's equation, and then the directrix is found using the standard form.

Option C ($\ln =a m^2$) is correct because it represents the condition for the line $lx + my + n = 0$ to be tangent to the parabola $y^2 = 4ax$.

Option C ($x = 0$) is correct because the directrix of the new parabola formed by the midpoints is $x=0$.

Option D ($24 / 5$) is correct. The length of the latus rectum is calculated using the given information about the focal chord segments.

Option B ($\sqrt 5$) is correct because the shortest distance between the parabolas occurs along the common normal, and the calculated distance is $\sqrt 5$.

Option B ($x + y + 1 = 0$) is correct because it represents the tangent to the parabola at the given point.

Option B (Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for statement-1) is correct because the line intersects the parabola at only one point, and the general tangent equation is correctly stated.

Option D (5) is correct because we find the common tangent to both curves, then calculate the distance from the given point to that line.

Option D ($\frac{{\sqrt 5 }}{2}$) is correct because it represents the shortest distance between the point and the curve, found by minimizing the distance function.

Option A ($\left(\frac{a}{m^2}, \frac{2 a}{m}\right)$) is correct because it represents the point of contact when the line y = mx + c is tangent to the parabola y² = 4ax.

Option C (8) is correct because the length of the subtangent to the parabola at a point is twice the abscissa of the point.

Option C (${m^3}y = 2a{m^2} - {m^2}x + a$) is correct because it represents the equation of the normal to the parabola at the given point, derived using the standard formula.

Option B (P_{1}, P_{2}, P_{3} are in G.P) is correct because the lengths of the perpendiculars from P, T, and Q to any tangent form a geometric progression.

Option D ($xy = \frac{{105}}{{64}}$) is correct because it represents the locus of the vertices of the given family of parabolas.

Option B ((-a, 0)) is correct because the chord AB always passes through the fixed point (-a, 0).

Option B $((-\infty, 1/3))$ is correct because it represents the set of values for $\lambda$ for which the line intersects the parabola at two distinct points.

Option A (Have only two common tangents which are mutually perpendicular) is correct because the common tangents are found to be $y = x + c$ and $y = -x + c$, which are perpendicular.

Option C (${(x + 1)}^2 = 4(y - 1)$) is correct because it represents the mirror image of the given parabola after reflection in the tangent line.

Option A ($\left\{ { - 1,1} \right\}$) is correct because these are the possible slopes of the focal chord that is tangent to the given circle.

Option A ($x=\frac{{3}}{{4}}$) is correct because the axis of the parabola is found by completing the square and identifying the x-coordinate of the vertex.

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

Option C ($x+y+1=0$) is correct because it is the equation of the tangent to the parabola $y^2=4x$ at the point (1, -2).

Option D (${90^0}$) is correct because the normals at the ends of the latus rectum of a parabola are always perpendicular.

Option D ($y=x-1$) is correct because it represents the tangent line to the parabola at x=1, found by calculating the derivative and using the point-slope form.

Option D ((18,-12)) is correct because it represents the coordinates of the point where the normal to the parabola at (2,4) meets the parabola again, calculated using the properties of the parabola and normal.

Option A (48) is correct because it represents the area of the triangle formed by the points of tangency and the focus of the parabola.

Option B ($\pi / 3$) is correct because the angle between the tangents is found by solving a quadratic equation and using the tangent formula.

Option C ($\frac{16}{25}$) is correct because it is the value of k that satisfies the given conditions for the intersection points of the two parabolas and the length of the line segment AB.

Option D (${y^2} = 2x$) is correct because the locus of the midpoint of a chord passing through the vertex of the parabola ${y^2} = 4x$ is ${y^2} = 2x$.

Option D (hyperbola) is correct because the locus of the point of contact is found to be $xy = 4a^2$, which represents a hyperbola.