Parabola practice · Mixed difficulty

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

Option C ((0,1)) is correct because it represents the point of contact when the straight line $x+y=1$ touches the parabola $y^2-y+x=0$.

Option C ((-4,4$\sqrt{2}$)) is correct because using the focal chord property of the parabola, the other extremity is found to be (-4,4$\sqrt{2}$).

Option A (Always in AP) is correct because the ordinates of the points A, C, and B on the parabola satisfy the condition for being in arithmetic progression.

Option C ($\pi / 3$) is correct because the angle between the tangents is found using the formula involving the slopes of the tangents and solving the quadratic equation.

Option C (${x^2} - 6x - 8y + 25 = 0$) is correct because it represents a parabola with vertex at (3,2) and focus at (3,4), derived from the given information.

Option D (Contains exactly two elements) is correct because solving the equation derived from the given conditions yields two real solutions for x.

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 B ($4 a \tan \alpha \sec \alpha$) is correct because it represents the length of the chord calculated using trigonometric relationships and the parabola's properties.

The correct answer is D ($9x - 4y + 4 = 0$) because it represents the tangent to the parabola that passes through the given point (4,10).

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 B (${x^2} - 4x + 8y + 12 = 0$) is correct because it represents the equation of the parabola with the given vertex and focus, derived using the definition of a parabola.

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 ((-2,11 / 2)) is correct because by completing the square, the equation can be rewritten in standard form, revealing the vertex.

Option C ($\left( {\frac{5}{4},1} \right)$) is correct because it represents the focus of the given parabola after rewriting the equation in standard form.

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 B ($3 \sqrt{2}$) is correct because the latus rectum is twice the distance between the focus and the directrix.

Option B (${y_1},{y_3},{y_2}$ are in AP) is correct because the ordinate of the intersection point of tangents to a parabola at two points is the arithmetic mean of the ordinates of those points.

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 A (5) is correct because the locus of the midpoints of parallel chords of a parabola is a line parallel to the axis of the parabola, and comparing with the given equation yields k=5.

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 (The line through Q parallel to y-axis) is correct because the tangents at P and R intersect on a vertical line passing through Q, given that the ordinates are in geometric progression.

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 C ($\frac{3}{2} \sqrt{41}$) is correct because it is the calculated length of the chord of contact using the given point and parabola equation.

Option B ((1,2) and (4,4)) is correct because these are the points of contact of the tangents from P(2,3) to the parabola y²=4x.

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 B (9) is correct because it is the value of 'k' when the given line is normal to the parabola.

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.