Perpendicular Distance of a Point from a Line
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Q.
A person standing at the junction of two straight paths represented by the equations and wants to reach the path whose equation is in the least time. Find the equation of the path that he should follow.
Q. The equation of a tangent to the parabola y2=8x which makes an angle 45∘ with the line y=3x+5 is
- 2x+y+1=0
- y=2x+1
- x+2y+8=0
- x−2y+8=0
Q.
The equation of tangent at on the curve is
Q. The distance between the lines 3x−4y+9=0 and 6x−8y−15=0 is ___ units
- 2033
- 1033
- 3310
- 3320
Q.
If the coordinates of a variable point be where is a variable quantity , find the locus of
Q. The set of points on the axis of the parabola y2−2y−4x+5=0 from which all the three normals to the parabola are real is :
- {(x, 1);x>3}
- {(x, 1);x≥3}
- {(x, 3);x≥1}
- {(x, −3);x>3}
Q. The locus of the point of intersection of perpendicular tangent drawn to each one of the parabola y2=4x+4 and y2=8x+16 is
- x=−3
- x=−12
- x=−8
- x=−4
Q. The condition that the straight line lx+my+n=0 touches the parabola x2=4ay is
- bn=am2
- al2−mn=0
- ln=am2
- am=ln2
Q. Let √3^i+^j, ^i+√3^j and β^i+(1−β)^j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is 3√2, then the sum of all possible values of β is :
- 1
- 2
- 3
- 4
Q. Equation(s) of the line perpendicular to x=y and at a distance of 1 unit from origin is/are
- x+y+√2=0
- 3x+4y+5=0
- x+y−√2=0
- 3x+4y−5=0
Q. If the line y−√3x+3=0 cuts the curve y2=x+2 at A and B, P is a point on the line whose ordinate is 0. Then |PA.PB|=
- 43(√3+2)
- 43(2−√3)
- 23(√3+2)
- 23(2−√3)
Q. The number of mutually perpendicular tangents that can be drawn from the curve y=||1−ex|−2| to the parabola x2=−4y is
Q. The equation of perpendicular bisectors of sides AB, BC of ΔABC are x−y−5=0 and x+2y=0 respectively. If A≡(1, −2), C≡(α, β), then α+β is equal to
Q. Let L be a line obtained from the intersection of two planes x+2y+z=6 and y+2z=4. If point P(α, β, γ) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21(α+β+γ) equals:
- 102
- 68
- 142
- 136
Q. The equation of tangent to the parabola y2=6x at point (6, 6) is
- 2x+3y−30=0
- 2x−y−6=0
- 2y−x−6=0
- x+y−12=0
Q. If the foot of the perpendicular from point (4, 3, 8) on the line L1:x−al=y−23=z−b4,
l≠0 is (3, 5, 7), then the shortest distance between the line L1 and line L2:x−23=y−44=z−55 is equal to :
l≠0 is (3, 5, 7), then the shortest distance between the line L1 and line L2:x−23=y−44=z−55 is equal to :
- 1√3
- 1√6
- √23
- 12
Q. The point of intersection of two tangents at the ends of the latus rectum to the parabola (y+3)2=8(x−2) is
- (0, −4)
- (0, −3)
- (−1, −3)
- (−2, −3)
Q.
The coordinates of a point at unit distance from the lines 3x - 4y + 1 = 0 and 3x + 6y + 1 = 0 are
(65, −110)
(0, −110)
(−25, −1310)
(−85, 310)
Q. One vertex of a square ABCD is A(−1, 1) and the equation of diagonal BD is 3x+y−8=0. Then
- C≡(−7, −1)
- C≡(5, 3)
- Area of square=40 sq. units
- Area of square=20 sq. units
Q. The length of perpendicular from the focus S of the parabola y2=4ax on the tangent at any point P on the parabola is
(O is the origin)
(O is the origin)
- √OS⋅SP
- OS⋅SP
- OS+SP
- OSOP
Q. The equation of circle which passes through focus of parabola x2=4y and touches it at (6, 9) is
- x2+y2+18x−28y+27=0
- x2+y2+24x−27y+26=0
- x2+y2+48x−12y+11=0
- x2+y2+18x−22y+21=0
Q. In ¯r.¯n=d, |d| represents the distance between
- Origin and any point on the plane
- Points where the plane intersects the axis
- Foot of perpendicular along ^n and origin
- None of these
Q. One vertex of a square ABCD is A(−1, 1) and the equation of diagonal BD is 3x+y−8=0. Then
- C≡(−7, −1)
- C≡(5, 3)
- Area of square=40 sq. units
- Area of square=20 sq. units
Q. If two tangents to y2=4ax make angles θ1, θ2 with positive x− axis such that cosθ1⋅cosθ2=k, then the locus of their point of intersection is
- x2=k2[(x−a)2+y2]
- x2=k2[(x+a)2+y2]
- x2=k2[(x−a)2−y2]
- x2=[(x+a)2+y2]
Q. A line passing through A(1, 2) is perpendicular to the lines 3x+4y−2=0 and 3x+4y+7=0 and intersecting them respectively at the points P and Q, then AQAP is equal to
Q. The point of intersection of tangents at the points on the parabola y2=4x whose ordinates are 4 and 6 is
- (6, 5)
- (7, 3)
- (9, 10)
- (6, 10)
Q. A variable line passes through a fixed point P, The algebraic sum of the perpendicular distances from (2, 0), (0, 2) and (1, 1) to the line is zero, then the coordinates of the P are
- (1, -1)
- (1, 1)
- (2, 1)
- (2, 2)
Q. If variable parameter α, β, γ∈R and satisfy the relations α2+2β2+3α=1+γ2 and 2α2+4β2=2γ2+5β, then
- Minimum value of α2+β2 is 461
- Minimum value of α2+β2 is 217
- Minimum value of α2+β2−2α+1 is 1661
- Minimum value of α2+β2−2α+1 is 117
Q. The equation of tangent to the parabola y2=6x at point (6, 6) is
- 2x+3y−30=0
- 2x−y−6=0
- 2y−x−6=0
- x+y−12=0
Q. Equation(s) of the common tangent to the parabola y2=24x and the circle x2+y2=18 is/are
- 2x+y+3=0
- x−y+6=0
- x+y+6=0
- x−2y+8=0